[e:main,bend left=20,->,color=BostonUniversityRed,line width=1.3pt] (bv83) to (v93);
[e:main,bend left=20,<-,color=AO,line width=1.3pt] (bs85) to (v95);
[e:main,bend left=20,->,color=BostonUniversityRed,line width=1.3pt] (bv87) to (v97);

[e:main,<-,color=AO,line width=1.3pt] (bs11) to (bs12);
[e:main,->,color=Gray] (bs12) to (bv13);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv13) to (bv14);
[e:main,->,color=Gray] (bv14) to (bv05);
[e:main,<-,color=AO,line width=1.3pt] (bs15) to (bs16);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bs16) to (bv17);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv17) to (bv18);

[e:main,->,color=AO,line width=1.3pt] (bs21) to (bs22);
[e:main,->,color=Gray] (bv23) to (bs12);
(bf23) [v:ghost,position=180:0.7mm from bv23] ;
[e:main,<-,color=BostonUniversityRed,line width=2.6pt] (bf23) to (bv24);
[e:main,<-,color=AO,line width=1.3pt] (bv23) to (bv24);
[e:main,<-,color=AO,line width=1.3pt] (bv24) to (bs25);
(bfff26) [v:ghost,position=45:0.7mm from bs26] ;
[e:main,->,color=BostonUniversityRed,line width=2.6pt] (bs25) to (bfff26);
[e:main,->,color=AO,line width=1.3pt] (bs25) to (bs26);
[e:main,<-,color=Gray] (bs16) to (bv27);
[e:main,<-,color=BostonUniversityRed,line width=1.3pt] (bv27) to (bv28);

[e:main,<-,color=AO,line width=1.3pt] (bs31) to (bs32);
[e:main,->,color=Gray] (bs32) to (bv33);
(bf34) [v:ghost,position=0:0.7mm from bv34] ;
[e:main,<-,color=BostonUniversityRed,line width=2.6pt] (bf34) to (bv33);
[e:main,->,color=AO,line width=1.3pt] (bv33) to (bv34);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv34) to (bs25);
[e:main,<-,color=AO,line width=1.3pt] (bs35) to (bs36);
[e:main,->,color=Gray] (bs36) to (bv37);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv37) to (bv38);

[e:main,->,color=AO,line width=1.3pt] (bs41) to (bs42);
[e:main,<-,color=AO,line width=1.3pt] (bs32) to (bv43);
(bf43) [v:ghost,position=180:0.7mm from bv43] ;
[e:main,<-,color=BostonUniversityRed,line width=2.6pt] (bf43) to (bv44);
[e:main,<-,color=AO,line width=1.3pt] (bv43) to (bv44);
[e:main,<-,color=Gray] (bv44) to (bs45);
[e:main,->,color=AO,line width=1.3pt] (bs45) to (bs46);
[e:main,<-,color=Gray] (bs36) to (bv47);
[e:main,<-,color=BostonUniversityRed,line width=1.3pt] (bv47) to (bv48);

[e:main,<-,color=AO,line width=1.3pt] (bs51) to (bs52);
[e:main,->,color=AO,line width=1.3pt] (bs52) to (bv53);
(bf54) [v:ghost,position=0:0.7mm from bv54] ;
[e:main,<-,color=BostonUniversityRed,line width=2.6pt] (bf54) to (bv53);
[e:main,->,color=AO,line width=1.3pt] (bv53) to (bv54);
[e:main,->,color=Gray] (bv54) to (bs45);
[e:main,<-,color=AO,line width=1.3pt] (bs55) to (bs56);
[e:main,->,color=Gray] (bs56) to (bv57);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv57) to (bv58);

[e:main,->,color=AO,line width=1.3pt] (bs61) to (bs62);
[e:main,<-,color=Gray] (bs52) to (bv63);
(bf63) [v:ghost,position=180:0.7mm from bv63] ;
[e:main,<-,color=BostonUniversityRed,line width=2.6pt] (bf63) to (bv64);
[e:main,<-,color=AO,line width=1.3pt] (bv63) to (bv64);
[e:main,<-,color=BostonUniversityRed,line width=1.3pt] (bv64) to (bs65);
[e:main,->,color=AO,line width=1.3pt] (bs65) to (bs66);
[e:main,<-,color=Gray] (bs56) to (bv67);
[e:main,<-,color=BostonUniversityRed,line width=1.3pt] (bv67) to (bv68);

[e:main,<-,color=AO,line width=1.3pt] (bs71) to (bs72);
[e:main,->,color=Gray] (bs72) to (bv73);
(bf74) [v:ghost,position=0:0.7mm from bv74] ;
[e:main,<-,color=BostonUniversityRed,line width=2.6pt] (bf74) to (bv73);
[e:main,->,color=AO,line width=1.3pt] (bv73) to (bv74);
[e:main,->,color=AO,line width=1.3pt] (bv74) to (bs65);
(bfff75) [v:ghost,position=135:0.7mm from bs75] ;
[e:main,->,color=BostonUniversityRed,line width=2.6pt] (bs76) to (bfff75);
[e:main,<-,color=AO,line width=1.3pt] (bs75) to (bs76);
[e:main,->,color=Gray] (bs76) to (bv77);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv77) to (bv78);

[e:main,->,color=AO,line width=1.3pt] (bs81) to (bs82);
[e:main,<-,color=Gray] (bs72) to (bv83);
[e:main,<-,color=BostonUniversityRed,line width=1.3pt] (bv83) to (bv84);
[e:main,<-,color=Gray] (bv84) to (bs85);
[e:main,->,color=AO,line width=1.3pt] (bs85) to (bs86);
[e:main,<-,color=BostonUniversityRed,line width=1.3pt] (bs76) to (bv87);
[e:main,<-,color=BostonUniversityRed,line width=1.3pt] (bv87) to (bv88);

[e:main,<-,color=AO,line width=1.3pt] (bs12) to (bs22);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv14) to (bv24);
(bfff16) [v:ghost,position=45:0.7mm from bs16] ;
[e:main,->,color=BostonUniversityRed,line width=2.6pt] (bs26) to (bfff16);
[e:main,<-,color=AO,line width=1.3pt] (bs16) to (bs26);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv18) to (bv28);

[e:main,<-,color=AO,line width=1.3pt] (bs21) to (bs31);
(bff33) [v:ghost,position=270:0.7mm from bv33] ;
[e:main,->,color=BostonUniversityRed,line width=2.6pt] (bv23) to (bff33);
[e:main,->,color=AO,line width=1.3pt] (bv23) to (bv33);

[e:main,<-,color=AO,line width=1.3pt] (bs25) to (bs35);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv27) to (bv37);

[e:main,<-,color=AO,line width=1.3pt] (bs32) to (bs42);
(bff44) [v:ghost,position=270:0.7mm from bv44] ;
[e:main,->,color=BostonUniversityRed,line width=2.6pt] (bv34) to (bff44);
[e:main,->,color=AO,line width=1.3pt] (bv34) to (bv44);
[e:main,<-,color=AO,line width=1.3pt] (bs36) to (bs46);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv38) to (bv48);

[e:main,<-,color=AO,line width=1.3pt] (bs41) to (bs51);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv43) to (bv53);
[e:main,<-,color=AO,line width=1.3pt] (bs45) to (bs55);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv47) to (bv57);

[e:main,<-,color=AO,line width=1.3pt] (bs52) to (bs62);
(bff64) [v:ghost,position=270:0.7mm from bv64] ;
[e:main,->,color=BostonUniversityRed,line width=2.6pt] (bv54) to (bff64);
[e:main,->,color=AO,line width=1.3pt] (bv54) to (bv64);
[e:main,<-,color=AO,line width=1.3pt] (bs56) to (bs66);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv58) to (bv68);

[e:main,<-,color=AO,line width=1.3pt] (bs61) to (bs71);
(bff73) [v:ghost,position=270:0.7mm from bv73] ;
[e:main,->,color=BostonUniversityRed,line width=2.6pt] (bv63) to (bff73);
[e:main,->,color=AO,line width=1.3pt] (bv63) to (bv73);
(bfff65) [v:ghost,position=135:0.7mm from bs65] ;
[e:main,->,color=BostonUniversityRed,line width=2.6pt] (bs75) to (bfff65);
[e:main,<-,color=AO,line width=1.3pt] (bs65) to (bs75);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv67) to (bv77);

[e:main,<-,color=AO,line width=1.3pt] (s81) to (bs11);

[e:main,->,color=BostonUniversityRed,line width=1.3pt] (v83) to (bv13);

[e:main,<-,color=AO,line width=1.3pt] (s85) to (bs15);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (v87) to (bv17);

[e:main,<-,color=AO,line width=1.3pt] (bs72) to (bs82);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv74) to (bv84);
[e:main,<-,color=AO,line width=1.3pt] (bs76) to (bs86);
[e:main,->,color=BostonUniversityRed,line width=1.3pt] (bv78) to (bv88);

(Label1) [v:ghost,position=270:50mm from W1,align=center] A slice $W$ of width $4$
of a cylindrical $8$-wall.;
(Label2) [v:ghost,position=270:50mm from W2] $B\coloneqq\Split{W}$ with the canonical matching $M$.;
(Label3) [v:ghost,position=270:50mm from W3] $B$ with the mixed matching $\Mix{W}$.;
(Label4) [v:ghost,position=270:50mm from W4] $\AB{W}$ and the twin walls of $W$.;

The `flipping' of a cylindrical $4$-wall into two overlapping cylindrical $2$-walls that are oppositely oriented.

Let us briefly investigate the operation of switching the perfect matching $M$ along a horizontal cycle of a [def:matchingwall]matching wall and its effect on the resulting $M'$-direction.

Let $k\in\N$ be a positive integer and $W$ be a cylindrical $4k$ wall, as well as $W'$ be a [def:slice]slice of width $2k$ of $W$.
Let us number the vertical cycles $W'$ inherits from $W$ as $Q_1,\dots,Q_{2k}$ and let us write $\hat{P}_1,\dots,\hat{P}_{4k}$ for the subpaths of the vertical paths of $W$ that still are present in $W'$.

Then $B_{W'}\coloneqq\Split{W'}$ is a slice of width $k$ of a [def:matchingwall]matching $2k$-wall, and we may assume $M$ to be its canonical matching.
Hence $\DirM{B_{W'}}{M}=W'$.
For each $i\in[1,2k]$ let $C_i$ be the $M$-conformal cycle of $B_{W'}$ such that $\DirM{C_i}{M}=Q_i$.
We define the mixed matching of $B_{W'}$, denoted by $\Mix{W'}$, as

\begin{align*}
\Mix{W'}\coloneqq M\Delta \bigcup_{\substack{i\in[1,2k]\\i\text{ odd}}}\E{C_i}.
\end{align*}

Thus $\Mix{W'}$ is obtained from $M$ by switching $M$ along every second vertical cycle starting with the first.
Let us denote by $\AB{W'}$ the digraph $\DirM{B_{W'}}{\Mix{W'}}$.
We say that $\A{W'}$ and $\B{W'}$ are the twin walls of $W'$ as illustrated in <ref>.

For the construction of $\A{W'}$ and $\B{W'}$, $W'$ must necessarily be of even width.
However, $\AB{W'}$ can be constructed for any slice, so in slight abuse of notation we will use $\AB{W'}$ to describe the digraph obtained from $\Split{W'}$ by switching the canonical matching along every second vertical cycle starting with the first.

Note that $\AB{W'}$ contains two cylindrical $k$-walls, one using the now flipped versions of the odd $Q_i$, denoted by $\A{W'}$, and the other using the still intact even $Q_i$, which is denoted by $\B{W'}$.
Moreover, if we start out with our embedding of $W$ and keep this embedding during all transformations, the vertical cycles of $\A{W'}$ go towards the top, while the vertical cycles of $\B{W'}$ still go to the bottom.
See <ref> for an illustration.
Moreover, every path connecting one of the vertical cycles of $\CC{W'}$ to another one, for some $\CSymb\in\Set{\ASymb,\BSymb}$, must necessarily visit all vertical cycles of its twin that lie in between.

This construction allows us to move up- and downwards almost arbitrarily in a sufficiently large wall, a huge advantage ober the more rigid digraphic setting.
In a slice $W'$ of the cylindrical wall, every directed cycle must visit all of the vertical paths, but in $\AB{W'}$ we are able to find directed cycles locally, which means there are strongly connected subgraphs of $\AB{W'}$ which lie within some tiles of $W'$.

The Model of $\Bidirected{K_t}$
Let us describe how the construction of the $\Bidirected{K_t}$-butterfly minor works.
We utilise essentially the same construction that was used in [6] to obtain <ref>.

Consider the slice $\widetilde{U}_1$ of width $d_1$ from the triadic partition of $U$.
Partition $\widetilde{U}_1$ into $t+1$ slices of equal width named $U'_1,\dots,U'_{t+1}$.
The first $t$ of these slices will hold the roots of the $t$ vertices of $\Bidirected{K_t}$, where the root of the $i$th vertex will be contained in $U'_i$.
The root of the $i$th vertex will be a path $V_i$, subpath of some horizontal path of $W_i'$ from left to right, with $t$ incoming edges $e^i_j$ and $t$ outgoing edges $f^i_j$ in such a way that

* the edges $e^i_j$ and $f^i_j$, $j\in[1,t-1]$, belong to vertical cycles of $U_i$, $i\in[1,t]$,

* the heads of $e^i_1,e^i_2,\dots,e^i_t$ appear in the order listed when traversing along $V_i$,

* the tails of $f^i_1,f^i_2,\dots,f^i_t$ appear in the order listed when traversing along $V_i$, and

* the head of $e^i_t$ appears on $V_i$ before the tail of $f^i_1$ when traversing along $V_i$.

We also require the slices $U'_i$ to appear in the order $U_1',U_2',\dots,U_t'$ from left to right, and that all $V_i$ belong to the same horizontal path.
If we are able to find a family of pairwise disjoint paths, internally disjoint from the roots, such that for every pair of distinct $i,j\in[1,t]$ the head of $f^i_j$ is linked to the tail of $e^j_i$, then the union of these paths together with the roots can be seen to form a butterfly minor model of $\Bidirected{K_t}$.
Indeed, each root can be contracted into a single vertex by the butterfly minor relation.
If we do this for each of the $t$ roots, then the resulting digraph is a subdivision of $\Bidirected{K_t}$.
For an illustration of a root see <ref>.

(v1) [v:main] ;
(v2) [v:main,position=0:12mm from v1] ;
(v3) [v:main,position=0:12mm from v2] ;
(v4) [v:main,position=0:12mm from v3] ;
(v5) [v:main,position=0:12mm from v4] ;
(v6) [v:main,position=0:12mm from v5] ;
(v7) [v:main,position=0:12mm from v6] ;
(v8) [v:main,position=0:12mm from v7] ;

(e1) [v:main,position=90:12mm from v1] ;
(e1L) [v:ghost,position=180:4.5mm from e1] $e^i_1$;

(e2) [v:main,position=90:12mm from v2] ;
(e2L) [v:ghost,position=180:4.5mm from e2] $e^i_2$;

(e3) [v:main,position=90:12mm from v3] ;
(e3L) [v:ghost,position=180:4.5mm from e3] $e^i_3$;

(e4) [v:main,position=90:12mm from v4] ;
(e4L) [v:ghost,position=180:4.5mm from e4] $e^i_4$;

(f1) [v:main,position=270:12mm from v5] ;
(f1L) [v:ghost,position=0:4.5mm from f1] $f^i_1$;

(f2) [v:main,position=270:12mm from v6] ;
(f2L) [v:ghost,position=0:4.5mm from f2] $f^i_2$;

(f3) [v:main,position=270:12mm from v7] ;
(f3L) [v:ghost,position=0:4.5mm from f3] $f^i_3$;

(f4) [v:main,position=270:12mm from v8] ;
(f4L) [v:ghost,position=0:4.5mm from f4] $f^i_4$;

[e:main,->] (v1) to (v2);
[e:main,->] (v2) to (v3);
[e:main,->] (v3) to (v4);
[e:main,->] (v4) to (v5);
[e:main,->] (v5) to (v6);
[e:main,->] (v5) to (v7);
[e:main,->] (v7) to (v8);

[e:main,->] (e1) to (v1);
[e:main,->] (e2) to (v2);
[e:main,->] (e3) to (v3);
[e:main,->] (e4) to (v4);

[e:main,->] (v5) to (f1);
[e:main,->] (v6) to (f2);
[e:main,->] (v7) to (f3);
[e:main,->] (v8) to (f4);

The the root of the model of the $i$th vertex of $\Bidirected{K_4}$.

A Refinement of <Ref>

In this paragraph we aim to get rid of the second and third outcome of <ref>.
These outcomes produce a set of semi local long jumps, meaning that these long jumps originate in the same strip and have their other endpoint almost immediately to the left or the right of the tiles where they originate.
For the digraphic setting this poses a problem since this kind of crosses cannot be used to create a $\Bidirected{K_t}$-butterfly minor model.
Hence the bounded number of `cross rows` in the definition of barely flatness.
In [6] this is overcome by a different argument highly increasing the necessary functions.
However, In our case we may use the additional freedom of the matching setting and thus the idea of twin walls to enable us to use to gu up and down in the wall to make use of all these crosses.

There exist functions $f_w\colon\N\rightarrow\N$, $f_P\colon\N\rightarrow\N$, and $f_W\colon\N\rightarrow\N$ such that for every $t\in\N$ the following holds:

* $D$ be a digraph,

* $W$ be a cylindrical $3k$-wall with $k\geq\Fkt{f_W}{t}$ in $D$,

* $\Triade=\Brace{W,k,W_1,W_2,W_3,W^1,W^2,W^3}$ be the [def:triadicpartition]triadic partition of $W$, and

* [def:tiling]$\Tiling=\Tiling_{W,k,\Fkt{f_w}{t}.\xi,\xi'}$ for some $\xi,\xi'\in[1,\Fkt{f_w}{t}+1]$.

If there exists a subfamily $\Tiling'$ of $\Tiling$ and a family $\Jumps$ of pairwise disjoint directed paths in $D$ with the following properties:

* Every member of $\Jumps$ is internally disjoint from $W$ but has both endpoints on $W$,

* $\Abs{\Tiling'}=\Abs{\Jumps}=\Fkt{f_P}{t}$,

* for every $T_{\Fkt{\ColumnFunction}{p},\Fkt{\RowFunction}{q},\Fkt{f_w}{t}}\neq T_{\Fkt{\ColumnFunction}{p'},\Fkt{\RowFunction}{q'},\Fkt{f_w}{t}}\in\Tiling'$ we have $\max\Set{\Abs{p-p'},\Abs{q-q'}}\geq 2$,

* there exists a bijection $\Start\colon\Tiling'\rightarrow\Jumps$ ($\End\colon\Tiling'\rightarrow\Jumps$) such that the starting point (endpoint) of the path $\Fkt{\Start}{T}$ ($\Fkt{\End}{T}$) belongs to the centre of $T$,

* $\V{\Fkt{\Start}{T}}\cap\V{\Tiling'}$ ($\V{\Fkt{\End}{T}}\cap\V{\Tiling'}$) contains exactly the endpoint of $\Fkt{\Start}{T}$ ($\Fkt{\End}{T}$) where $\V{\Tiling'}=\bigcup_{T'\in\Tiling'}\V{T'}$, and finally

* the endpoints (starting points) of the paths in $\Jumps$ are of mutual $W$-distance at least $4$.

Then $\Split{D}$ has a $K_{t,t}$-[def:matchingminor]matching minor grasped by $\Split{W}$.

We may assume that case b) or c) of <ref> holds, as in case a) we are done by <ref>.
Moreover, we may assume to be in case b) since case c) can be handled analogously.
Let $S\subseteq W$ be the [def:strip]strip, $\mathcal{T}''\subseteq \mathcal{T}'$, $\Abs{\mathcal{T}''}\geq\Fkt{f_P}{t}^{\frac{1}{4}}$, be the tiles, and $\mathcal{J}'\subseteq\mathcal{J}$, $\Abs{\mathcal{J}'}=\Abs{\mathcal{T}''}$, be the family of jumps from case b).
For each tile $T\in\mathcal{T}''$ let us define $s(T)$ to be the [def:slice]slice of width $2\Fkt{f_P}{t}\leq \frac{1}{2}\Fkt{f_w}{t}$ such that each vertical path of $\Fkt{s}{T}$ intersects $T$ and $\Fkt{s}{T}$ contains the right most vertical path of $T$.
Please note that in $W-\Fkt{s}{T}$ there are still at least $2\Fkt{f_P}{t}$ vertical paths left that pass through $T$ to the right of the [def:tile]centre of $T$.
Let us also define $\Fkt{c}{T}$ to be the [def:slice]slice of $W$ of width $2\Fkt{f_P}{t}+2$ whose vertical paths all pass through $T$ and that has exactly $\Fkt{f_P}{t}+1$ vertical paths on either side of the centre of $T$.
So $\Fkt{c}{T}$ and $\Fkt{s}{T}$ are disjoint and whenever we have a family $\mathcal{P}$ of at most $\Fkt{f_P}{t}$ pairwise disjoint paths entering $T\cap\Fkt{c}{T}$ from the top we can select any path $P\in\mathcal{P}$ to be routed into the centre of $T$ while all other paths in $\mathcal{P}$ can be routed within $\Fkt{c}{T}$ to the bottom of $T$ while maintaining their disjointness.

Our goal is to make use of this property to achieve the following:
Suppose we have a family $\mathcal{P}$ of at most $\Fkt{f_P}{t}^{\frac{1}{4}}$ pairwise disjoint paths starting an the left perimeter of $W_2$.
We want to visit the tiles of $T\in \mathcal{T}''$ consecutively from left to right, each time sending some path of $\mathcal{P}$ into the [def:tile]centre of $T$ and then into the corresponding jump from $\mathcal{J}'$.
If we manage to do this in such a way that in the end each path in $\mathcal{P}$ is assigned some long jump in $\mathcal{J}'$ and we can find a linkage from the endpoints of the jumps back to the left perimeter of $W_2$ without intersecting our previous routing, then we can use this to create any digraph on $t$ vertices as a butterfly minor by positioning the roots of the corresponding model within $W_1$ and attaching their in and out paths to the end- and start points of our routing on the left perimeter of $W_2$.
Hence this is enough to proof the lemma.

However, this cannot be done within the wall $W$ as it is because all vertical paths go downwards.
This means, to reach the next tile after we have visited some tile $T\in\mathcal{T}''$ we need to go all the way down to the bottom of $\Fkt{s}{T}$ and then use the flying edges of $W$ to reach the top part again.
By doing so it might become impossible for any path on the right of $\Fkt{s}{T}$ to reach the left perimeter of $W_2$ without crossing through the previously constructed routing.
To circumvent this problem we turn each $\Fkt{s}{T}$ into its two [fig:flippingawall]twin walls.
This allows us to leave $\Fkt{c}{T}$ with our paths immediately after passing by the [def:tile]centre of $T$ and continuing to the right until all remaining paths have reached their position in $\AB{\Fkt{s}{T}}$.
Here the remaining paths can now be routed upwards within $\AB{\Fkt{s}{T}}$ to ensure they can enter the next tile from above, while its corresponding jump can be routed to the left without interference.
This strategy might change the perfect matching and thereby change the digraph $D$, however, we will still find a $\Bidirected{K_t}$-butterfly minor model grasped by the changed version of $W$.
Hence we will find a $K_{t,t}$-[def:matchingminor]matching minor grasped by [def:split]$\Split{W}$ as desired.

We now describe how this routing is constructed.
Let $k\leq\Fkt{f_P}{t}$ be some positive integer and let $\pi$ be some permutation of $[1,k]$.
Furthermore, let us select two additional strips $S_1$ and $S_2$ of $W_2$, both of the same height as $S$, such that $S$ separates $S_1$ from $S_2$ in $W_2$ and $S_1$ is above $S$.
Let $s_1,\dots,s_k$ be the starting points of $k$ left-to-right paths, let us call them $P_1,\dots,P_k$, of $S_1$ such that for each $i\in[1,k]$ $s_i$ is the starting point of $P_i$ and for each $i\in[1,k-2]$ the path $P_{i+1}$ separates $P_i$ from $P_{i+2}$.
Additionally, let us assume the tiles of $\mathcal{T}''=\Set{T_1,T_2,\dots,T_{\Abs{\mathcal{T}''}}}$ to be numbered from left to right as in outcome b) of <ref>.
We start with $T_1$.
Visiting $T_1$ is done as follows:
In $\Fkt{c}{T_1}\cap T_1$ select $k-\Fkt{\pi}{1}$ distinct vertical paths $Q_k,\dots,Q_{k-\Fkt{\pi}{1}}$ occurring from left to right in the order listed such that $Q_{k-\Fkt{\pi}{1}}$ does not intersect the boundary of the centre of $T_1$ but still lies left of it.
Let $Q_{k-\Fkt{\pi}{1}-2},\dots,Q_1$ be another family of distinct vertical paths of $\Fkt{c}{T_1}\cap T_1$ occurring from left to right in the order listed, being disjoint from the centre of $T_1$ and occurring to the right of the centre $T_1$
Finally let $Q_{k-\Fkt{\pi}{1}-1}$ let be a shortest path in $D'$ starting in the start-vertex of the vertical path of $T_1$ that passes through the centre of $T_1$ and ending in the starting-point of the path $\Fkt{\Start}{T_1}$, where $D'$ is the digraph obtained from the union of $T_1$ and the subgraph of $D$ that is separated from $W$ by the boundary of the centre of $T_1$.
We may now find a linkage from $\Set{s_1,\dots,s_k}$ to the starting points of the $Q_i$ such that for each $i\in[1,k]$ the vertex $s_i$ is linked to $Q_i$ and the linkage stays entirely in the quadrant of $W_2$ defined by the upper path of $S$ and the left most path of $\Fkt{s}{T_1}$.
Moreover, we may choose this linkage to stay as much as possible within the strip $S_1$.
We may now continue with each of the paths routed so far until for each $i\in[1,k]\setminus\Set{\Fkt{\pi}{1}}$ the path with starting point $s_i$ meets $P_i$ again.
Let $W_2^2$ be the largest slice of $W_2$ containing the right perimeter of $W_2$ while being disjoint from $T_1$.
Note that
From here on, each of these paths may traverse along its respective $P_i$ to the right until it meets a vertex of the left perimeter of $W_2^2$ for the first time.

Note that we may now find a bijection $\varphi_1\colon[1,k-1]\rightarrow[1,k]\setminus\Set{1}$ such that $\Fkt{\varphi}{i}=i+1$ and a permutation $\pi_1$ of $[1,k-1]$ such that $\Fkt{\pi_1}{i}=\Fkt{\pi}{\Fkt{\varphi_1}{i}}$ for all $i\in[1,k-1]$.
This allows us to proceed with the construction of this part of the routing via induction.
In the end, for each $i\in[1,k]$ we have sent the $\Fkt{\pi}{i}$th path from the centre of $T_i$ to the left.
Let now $L_1,\dots,L_k$ be $k$ distinct right-to-left paths in $S_2$ occurring from top to bottom in the order listed.
For each $i\in[1,k]$ we may now find a path from the endpoint of $\Fkt{\Start}{T_i}$ to $L_i$ by moving along a single vertical path of $W_2$.
It follows from the construction in the previous step that each of these paths is disjoint from all parts of the routing so far.
We may now complete the second step by enhancing each of the paths build so far by following along the $L_i$ to their respective endpoints on the left perimeter of $W_2$.
Let $t_1,\dots,t_k$ be these endpoints such that $t_i$ is the endpoint of $L_i$ for all $i\in[1,k]$.
We have now found $k$ pairwise disjoint paths $R_1,\dots,R_k$ such that for each $i\in[1,k]$ $R_i$ starts in $s_i$ and ends in $t_{\Fkt{\pi}{i}}$.

To complete the proof let us find $t$ pairwise disjoint [def:slice]slices of $W_1$, each of width $2\Fkt{f_w}{t}+2$ and let $U_1,\dots,U_t$ be these slices numbered according to their occurrence in $W_1$ from left to right.
For each $i\in[1,t]$ let us create a root $F_i$ of $\Bidirected{K_t}$ such that all roots together fit into a strip $S'$ of height $2\Fkt{f_w}{t}+2$ that can be combined with the strip $S$ to form a strip of $W_1\cup W_2$.
Moreover, let us create the root slightly different to how it was defined above:
Let the base path go from right to left, let the outgoing paths leave to the top on the left and let the incoming paths enter from the bottom on the right.
To make this possible we may create the $F_i$ in a way such that they are also all contained in a single slice of width $2\Fkt{f_w}{t}+2$ of $W_1$ and, moreover, such that the outgoing parts all belong to a subslice of with at most $\Fkt{f_w}{t}$.
For this subslice we switch the matching of the split of $W_1$ to form the twin wall.
Now the required upwards paths exist.
Let $S_1'$ be the strip of $W_1$ such that $S_1'\cup S_1$ is a strip of $W_1\cup W_2$.
We may now find a linkage from $\CondSet{f^j_i}{i\in[1,t]\text{, and }j\in[1,t]\setminus\Set{i}}$ to the set of starting points of left-to-right paths of $S_1$ on the left perimeter of $W_2$.
Similarly we may connect the vertices $t_i$ from above to the vertices in $\CondSet{e^j_i}{i\in[1,t]\text{, and }j\in[1,t]\setminus\Set{i}}$.
As proven above we may create any particular linkage of the $s_i$ to the $t_i$ within $W_2$ together with the jumps $\mathcal{J}'$ and thus we are able to construct a butterfly minor model of $\Bidirected{K_t}$ in some $M$-direction of $\Split{D}$.

Further refining <Ref> and removing long jumps

<Ref> is already a powerful tool.
However, it is not straightforward how to obtain the apex set $A$ just from its application.
Hence in the following, we aim for further refinement and the existence of the set $A$.

Let $f_w\colon\N\rightarrow\N$ be some function, $t,k\in\N$ two positive integers, and $\xi,\xi'\in[1,\Fkt{f_w}{t}+1]$.
Moreover, let $W=\Brace{Q_1,\dots,Q_{3k},\hat{P}_1,\dots,\hat{P}_{3k}}$ be a cylindrical $3k$-wall with its [def:triadicpartition]triadic partition $\Triade=\Brace{W,k,W_1,W_2,W_3,W^1,W^2,W^3}$ and [def:tiling]$\Tiling=\Tiling_{W,k,\Fkt{f_w}{t},\xi,\xi'}$.
A colouring of $\Tiling$ is a partition of $\Tiling$ into four classes, namely $\Class_1,\Class_2,\Class_3$, and $\Class_4$ as follows:
For every $i\in\left[1,\Ceil{\frac{k+\xi-1}{2\Fkt{f_w}{t}+1}}+1\right]$ and every $j\in \left[1,\Ceil{\frac{3k-\xi'-1}{2\Fkt{f_w}{t}+1}}+1\right]$ we assign to $T_{\Fkt{\ColumnFunction}{i},\Fkt{\RowFunction}{j},\Fkt{f_w}{t}}$ the colour $\Brace{i\mod 2}+2\Brace{j\mod 2}+1$.
This means that tiles where $\Fkt{\ColumnFunction}{i}$ and $\Fkt{\RowFunction}{j}$ are even get colour $1$, the tiles where $\Fkt{\RowFunction}{j}$ is even but $\Fkt{\ColumnFunction}{i}$ is odd get $3$, and so on.
Hence every column is two-chromatic, every row is so as well, and between each pair of tiles from the same colour that share a row or a column, there is a tile of a different colour that separates those tiles in their respective row or column.
Additionally, if $T$ is some tile, then the eight tiles surrounding $T$ are all of different colour than $T$ itself.

Let $t,k,k',w\in\N$ be positive integers such that $k\geq k'\geq2\Fkt{f_W}{t}+4\Fkt{f_P}{t}\Brace{2w+1}$, $w\geq 2\Fkt{f_w}{t}$, and $\xi,\xi'\in\left[1,w+1\right]$.
Let $D$ be a digraph containing a cylindrical $3k$-wall $W=\Brace{Q_1,\dots,Q_{3k},\hat{P}_1,\dots,\hat{P}_{3k}}$ with its [def:triadicpartition]triadic partition $\Triade=\Brace{W,k,W_1,W_2,W_3,W^1,W^2,W^3}$, and a tiling [def:tiling]$\Tiling=\Tiling_{W,k,w,\xi,\xi'}$.
Let $i\in[1,4]$, $\Set{\Class_1,\dots,\Class_4}$ be a four colouring of $\Tiling$ and $W'\subseteq W$ be a [def:slice]slice of width $k'$ of $W_2$.
At last, let us denote by $\Tiling'$ the family of tiles from $\Tiling$ that share a vertex with $W'$.
Similarly let $\Class_i'\coloneqq\Tiling'\cap\Class_i$.
Then $\Fkt{D^1_i}{W'}$ is the digraph obtained from $D$ by performing the following construction steps for every $T\in\Class_i'$:

* add new vertices $x_T^{\text{in}}$ and $x_T^{\text{out}}$,

* for every vertex $u$ in the centre of $T$ introduce the edges $\Brace{u,x_T^{\text{in}}}$ and $\Brace{x_T^{\text{out}},u}$, and then

* delete all internal vertices of $T$.

We also need the following result:
Let $D$ be a digraph and $X,Y\subseteq\V{D}$. A half-integral $X$-$Y$-linkage of order $k$ is a family $\mathcal{P}$ of directed $X$-$Y$-paths such that every vertex of $D$ is contained in at most two paths from $\mathcal{P}$.
By $\V{\mathcal{P}}$ we denote the set $\bigcup_{P\in\mathcal{P}}\V{P}$.

Let $k\in\N$ be a positive integer, $D$ be a digraph, and $X,Y\subseteq\V{D}$.
If $\mathcal{P}$ is a half-integral $X$-$Y$-linkage of order $2k$ in $D$, then there exists a family $\mathcal{J}$ of pairwise disjoint $X$-$Y$-paths such that $\V{\mathcal{J}}\subseteq\V{\mathcal{P}}$.

Let $t,k,k',w\in\N$ be positive integers such that $k\geq k'\geq2\Fkt{f_W}{t}+2^{16}\Fkt{f_P}{t}+2$, $w\geq 2\Fkt{f_w}{t}+2^7\Fkt{f_P}{t}$, and $\xi,\xi'\in\left[1,w+1\right]$.
Let $D$ be a digraph containing a cylindrical $3k$-wall $W=\Brace{Q_1,\dots,Q_{3k},\hat{P}_1,\dots,\hat{P}_{3k}}$ with its [def:triadicpartition]triadic partition $\Triade=\Brace{W,k,W_1,W_2,W_3,W^1,W^2,W^3}$, and a tiling [def:tiling]$\Tiling=\Tiling_{W,k,w,\xi,\xi'}$.
Let $i\in[1,4]$, $\Set{\Class_1,\dots,\Class_4}$ be a four colouring of $\Tiling$ and $W'\subseteq W$ be a [def:slice]slice of width $k'$ of $W_2$.

Now let $\Tiling'$ be the family of all tiles of $\Tiling$ that are completely contained in $W'$ and let $\widetilde{W}$ be the smallest slice of $W$ that contains all tiles from $\Tiling'$.

Consider the [def:auxiliarydigraphI]auxiliary digraph of type I $\Fkt{D_i^1}{\widetilde{W}}$ and let $\Class_i'$ be as in the definition of $\Fkt{D_i^1}{\widetilde{W}}$.
Define the sets

\begin{align*}
X_{\text{I}}^{\text{out}}&\coloneqq\CondSet{x_T^{\text{out}}}{T\in\Class_i'}\text{, and}\\
\end{align*}

Additionally, we construct the set $Y_{\text{I}}$ as follows:
Let $Q$ and $Q'$ be the two cycles of $\Perimeter{W_2}$.
For every $j\in\left[ 1,\frac{3k}{4} \right]$, $Y_{\text{I}}$ contains exactly one vertex of $Q\cap P^1_{4j}$, $Q\cap P^2_{4j+2}$, $Q'\cap P^1_{4j}$, and $Q'\cap P^2_{4j+2}$ each.

If there exists a family $\mathcal{L}$ of pairwise disjoint directed paths with $\Abs{\mathcal{L}}= 2^7\Fkt{f_P}{t}$ such that either

* $\mathcal{L}$ is a family of directed $X_{\text{I}}^{\text{out}}$-$Y_{\text{I}}$-paths, or

* $\mathcal{L}$ is a family of directed $Y_{\text{I}}$-$X_{\text{I}}^{\text{in}}$-paths,

then $\Split{D}$ has a $K_{t,t}$-[def:matchingminor]matching minor grasped by [def:split]$\Split{W}$.

The proof is divided into several steps and we start with a brief outline.
Our goal is to construct a cylindrical wall $U\subseteq W$ of sufficient size, together with a family of $\Fkt{f_P}{t}$ directed $U$-paths that meet the requirements of <ref>.
If we are able to do this, then <ref> yields the desired outcome.

Without loss of generality, let us assume $\mathcal{L}$ is a family of directed $X_{\text{I}}^{\text{out}}$-$Y_{\text{I}}$-paths.
The other case can be seen using similar arguments.

Let $L$ and $P$ be directed paths.
We say that $P$ is a long jump of $L$ if $P$ is a $w$-long jump over $W$ and $P\subseteq L$.
We also say that $P$ is a jump of $L$ if $P$ is a directed $W$-path.

Towards our goal, we first show that we can use $\mathcal{L}$ to construct a half-integral $X_{\text{I}}^{\text{out}}$-$Y_{\text{I}}$-linking $\mathcal{L}_1$ such that

* $\Abs{\mathcal{L}_1}=2^7\Fkt{f_P}{t}$,

* there exists a family $\mathcal{F}\subseteq\Tiling'$ with $\Abs{\mathcal{F}}\leq 2^7\Fkt{f_P}{t}$, and

* for every $L\in\mathcal{L}_1$, every endpoint $u$ of a jump of $L$ with $u\in\V{\widetilde{W}}$ belongs to a tile from $\Class_i'\cup\mathcal{F}$.

Once this is achieved, we use <ref> to obtain a family $\mathcal{L}_2$ of pairwise disjoint directed $X_{\text{I}}^{\text{out}}$-$Y_{\text{I}}$-paths of size $2^6\Fkt{f_P}{t}$ from $\mathcal{L}_1$.
Afterwards, we remove the cycles and paths of $\widetilde{W}$ that meet tiles from $\mathcal{F}$ and obtain a new slice $\widetilde{W}'$ of some cylindrical wall.
For this slice, we construct a [def:tiling]tiling and a [def:tilingII]tier II tiling as well as a half-integral linking $\mathcal{L}_4$ of size $2^6\Fkt{f_P}{t}$ from $\mathcal{L}_3$ that connects the [def:tile]centres of some tiles in the tier II tiling to vertices of $\widetilde{W}'$ such that their endpoints are mutually far enough apart and every path in $\mathcal{L}_4$ is internally disjoint from the new wall.
Another application of <ref> then yields the family of long jumps necessary for an application of <ref>.

We start out with the construction of $\mathcal{L}_1$ and $\mathcal{F}$.
For this, let $\mathcal{L}'\coloneqq \mathcal{L}$, $\mathcal{L}_1\coloneqq \emptyset$, and $\mathcal{F}\coloneqq\emptyset$.
As long as $\mathcal{L}'$ is non-empty, perform the following actions:

Select some path $L\in\mathcal{L}$.
In case $L$ is internally disjoint from $\widetilde{W}$, add $L$ to $\mathcal{L}_1$ and remove it from $\mathcal{L}'$.
Otherwise let $s_L$ be its starting point and let $v_L$ be the first vertex of $L$, when traversing along $L$ starting from $s_L$, that belongs to $\widetilde{W}$, but not to a tile from $\Class_i'$.

* If $v_L$ does not belong to a tile from $\mathcal{F}$, let $T\in\Tiling\setminus\Class_i$ be the tile that contains $v_L$ and add $T$ to $\mathcal{F}$.
Let $R$ be a shortest directed path from $v_L$ to $Y_{\text{I}}$ in $W$ such that $R$ avoids all vertices of $W$ that are contained in two different paths of $\mathcal{L}_1$ and that is internally disjoint from $Lv_L$.
Now add $Lv_LR$ to $\mathcal{L}_1$ and remove $L$ from $\mathcal{L}'$.
Note that such a path $R$ must exist since the paths in $\mathcal{L}$ are pairwise disjoint, we never used $T$ for such a re-routing before, and $w$ and $k'$ are chosen sufficiently large in proportion to $2^7\Fkt{f_P}{t}$.
Also note that the path $R$ is exactly the part, where we might go from integral to half-integral, but since our paths were pairwise disjoint to begin with, we can be sure that $R$ does never meet a vertex contained in two distinct paths.

* So now suppose $v_L$ belongs to a tile $T$ from $\mathcal{F}$.
Let us follow along $v_LL$ until the first time we encounter a vertex $u_L$ for which one of the following is true:

* $u_L$ belongs to a tile $T$ from $\Tiling\setminus\Brace{\Class_i\cup\mathcal{F}}$, or

* every internal vertex of $u_LL$ belongs to $W-\widetilde{W}$ or to some tile from $\Class_i\cup\mathcal{F}$.

If a) is the case, repeat the instruction from i) but replace $v_L$ by $u_L$.
In this case $T$ is added to $\mathcal{F}$.
Otherwise b) must hold and here we may simply remove $L$ from $\mathcal{L}'$ and add it to $\mathcal{L}_1$.

Now for every $L\in\mathcal{L}$ we added at most one tile to $\mathcal{F}$ and thus $\Abs{\mathcal{F}}\leq\Abs{\mathcal{L}}$.
Moreover, from the construction it is clear that $\mathcal{L}_1$ is indeed a half-integral linkage from $X_{\text{I}}^{\text{out}}$ to $Y_{\text{I}}$.
Also, please note that we may assume that every $L$ meets each tile in $\mathcal{F}$ in at most $2^7\Fkt{f_P}{t}+1$ horizontal path pairs and vertical cycles, since otherwise one could find a short cut through $W$ itself.

Next, we may apply <ref> to obtain a family $\mathcal{L}_2$ of pairwise disjoint directed $X_{\text{I}}^{\text{out}}$-$Y_{\text{I}}$-paths with $\V{\mathcal{L}_2}\subseteq\V{\mathcal{L}_1}$ and $\Abs{\mathcal{L}_2}=2^6\Fkt{f_P}{t}$.
This completes the second step.

For the third step, let us consider $W''\coloneqq \InducedSubgraph{\widetilde{W}}{\Tiling,i}$ together with the tiling $\Tiling''\coloneqq\TierIITiling{\Tiling,i,f}{\widetilde{W}}$ and a four-colouring $\Set{\widetilde{\Class}_1,\dots,\widetilde{\Class}_4}$.
Note that by choice of $k'$ this means that $W''$ is a slice of width $k''\geq1\Fkt{f_W}{t}+2^7\Fkt{f_P}{t}\Brace{2w+1}+1$ of some cylindrical $3k''$-wall that is completely contained in $W$.
For each $L\in\mathcal{L}_2$ let $T^1_L\in \Class_i$ such that the starting point $s_L$ of $L$ belongs to $T^1_L$.
Let $K^1_L\in \Tiling''$ be the tile whose centre is the perimeter of $T^1_L$.
Choose any vertex $s'_L$ of degree three in $W''$ that is not contained in any path of $\mathcal{L}_2$, and let $R_L$ be a directed path from $s_L$ to $s'_L$ within $T^1_L$.
Let $\mathcal{L}_3'$ be the resulting, and potentially now again half-integral, family of directed paths.
Now there must exist $j\in[1,4]$ such that at least $2^4\Fkt{f_P}{t}$ of the paths from $\mathcal{L}_3'$ start at the centre of a tile from $\widetilde{\Class}_j$.
Let $\mathcal{L}_3''\subseteq\mathcal{L}_3'$ be a family of exactly $2^4\Fkt{f_P}{t}$ such paths.
Next let us consider the family $\mathcal{F}$.
Let $W'''$ be the subgraph of $W''$ induced by all vertical cycles and horizontal path pairs in $W''$ that do not contain a vertex of some tile in $\mathcal{F}$ that belongs to a path in $\mathcal{L}_3''$.
Since $\Abs{\mathcal{F}}\leq 2^7\Fkt{f_P}{t}$ and each tile in $\mathcal{F}$ meets a path in $\mathcal{L}_3''$ in at most $2^7\Fkt{f_P}{t}+1$ such cycles and pairs of horizontal paths, it follows that $W'''$ is a slice of width $k'''\geq\Fkt{f_W}{t}+2$ of some cylindrical $3k'''$wall $W^*\subseteq W$.
Moreover, $W^*$ can be partitioned into three [def:slice]slices of width $k'''$ as in its [def:triadicpartition]triadic partition, such that $W'''$ is the slice in the middle.
Let us rename the paths and cycles of $W^*$ such that $W^*=\Brace{Q^*_1,\dots,Q^*_{3k'''},\hat{P^*}_1,\dots,\hat{P^*}_{3k'''}}$, and we construct the set $Y^*$ as follows:
Let $Q^*$ and $'Q^*$ be the two cycles of $\Perimeter{W'''}$.
For every $j\in\left[ 1,\frac{3k'''}{4} \right]$, $Y^*$ contains exactly one vertex of $Q^*\cap P^{*1}_{4j}$, $Q\cap P^{*2}_{4j+2}$, $'Q^*\cap P^{*1}_{4j}$, and $'Q^*\cap P^{*2}_{4j+2}$ each.
Let $L\in\mathcal{L}_3''$ be any path and $t_L$ be the first vertex after its starting point $L$ shares with either $W'''$ or $W^*-W'''$.
In case $t_L\in\V{W'''}$, simply add $Lt_L$ to $\mathcal{L}_3'''$.
Otherwise, let $b_L$ be the endpoint of $L$ in $W^*-W'''$.
Then we can find a path $R_L$ in $W$ from $b_L$ to a vertex $t^*_L$ of $Y^*$ such that $t^*_L$ is of $W^*$-distance at least $4$ to every endpoint of every path already in $\mathcal{L}_3'''$, $R_L$ is internally disjoint from $L$, and $R_L$ does not contain a vertex that is contained in two distinct paths from $\mathcal{L}_3'''$.
Add $LR_L$ to $\mathcal{L}_3'''$.
Finally, $\mathcal{L}_3'''$ is a half-integral linkage from the set starting points $S^*$ of the paths in $\mathcal{L}_3''$ to $Y^*$ of size $2^4\Fkt{f_P}{t}$, and thus by <ref> we can find a family $\mathcal{L}_4$ of pairwise disjoint directed paths from $S^*$ to $Y^*$ with $\V{\mathcal{L}_4}\subseteq\V{\mathcal{L}_3'''}$ that is of size $2^3\Fkt{f_P}{t}$.
It follows that all paths in $\mathcal{L}_4$ are internally disjoint from $W'''$.
This concludes the fourth step.

Let us consider the tiles of $\widetilde{\Class}_i$ whose centres contain a vertex of $S^*$.
Since $W'''$ might be a proper subgraph of $W''$, $\Tiling''$ is not necessarily a tiling of $W'''$.
Each such tile $T$, however, contains a tile $T'$ of width $\Fkt{f_w}{t}$ with the same [def:tile]centre.
Since $T$ can be surrounded by at most $8$ tiles from $\mathcal{F}$ in $W'$, we may find, among the $2^3\Fkt{f_P}{t}$ many such tiles, a family $\Jumps$ of $\Fkt{f_P}{t}$ tiles that are pairwise disjoint and thus, since they all are constrcuted from the family $\widetilde{\Class}_i$, they meet the distance requirements of the tiles in <ref>.
Hence we may apply <ref> and obtain a
$K_{t,t}$-[def:matchingminor]matching minor grasped by [def:split]$\Split{W^*}$ $\Bidirected{K_t}$-butterfly minor grasped by $W^*$.
Moreover, since $W^*\subseteq W$, this completes the proof of our lemma.

With this we are able to handle all long jumps that attach to tiles of different colour.
Using a second auxiliary digraph, we find a similar way to handle those long jumps over $W$ that attach to tiles of the same colour by using <ref>.

Let $t,k,k',w\in\N$ be positive integers such that $k\geq k'\geq2\Fkt{f_W}{t}$, $w\geq 2\Fkt{f_w}{t}$, and $\xi,\xi'\in\left[1,w+1\right]$.
Let $D$ be a digraph containing a cylindrical $3k$-wall $W=\Brace{Q_1,\dots,Q_{3k},\hat{P}_1,\dots,\hat{P}_{3k}}$ with its [def:triadicpartition]triadic partition $\Triade=\Brace{W,k,W_1,W_2,W_3,W^1,W^2,W^3}$, and a [def:tiling]tiling $\Tiling=\Tiling_{W,k,w,\xi,\xi'}$.
Let $i\in[1,4]$, $\Set{\Class_1,\dots,\Class_4}$ be a four colouring of $\Tiling$ and $W'\subseteq W$ be a [def:slice]slice of width $k'$ of $W_2$ such that no [def:tile]tile of $\Class_i$ contains a vertex of the perimeter of $W'$.
Then $\Fkt{D^2_i}{W'}$ is the digraph obtained from $D$ by performing the following construction steps:

for every $T\in\Class_i$, such that $T$ contains a vertex of $W'$, we do the following:

* add a new vertex $x_T$, and

* for every vertex $v$ that belongs to the interior or the centre of $T$, introduce the edges $\Brace{x_T,v}$ and $\Brace{v,x_T}$.

Once this is done, delete all vertices of $W'$ that do not belong to tiles of $\Class_i'$
Let $X_{\text{II}}^i$ be the collection of all newly introduced vertices $x_T$.

Let $t,k,k',w\in\N$ be positive integers, and $\xi,\xi'\in\left[1,w+1\right]$ where $w\geq2\Fkt{f_w}{t}$.
Let $D$ be a digraph containing a cylindrical $3k$-wall $W_0=\Brace{Q_1,\dots,Q_{3k},\hat{P}_1,\dots,\hat{P}_{3k}}$, where $k\geq k'\geq4\Fkt{f_W}{t}^2$, with its [def:triadicpartition]triadic partition $\Triade=\Brace{W_0,k,W_1,W_2,W_3,W^1,W^2,W^3}$, a [def:slice]slice $W\subseteq W_2$ of width $k'$, a [def:tiling]tiling $\Tiling=\Tiling_{W,k,w,\xi,\xi'}$, a four colouring $\Set{\Class_1,\dots,\Class_4}$, and a fixed colour $i\in[1,4]$.

Then $D$ has a $K_{t,t}$-[def:matchingminor]matching minor grasped by [def:split]$\Split{W_0}$, or there exists a set $\MarkedTiles{2}_{i,\xi,\xi'}\subseteq\Tiling$ with $\Abs{\MarkedTiles{2}_{i,\xi,\xi'}}\leq 8\Fkt{f_P}{t}$ and a set $Z^2_{i,\xi,\xi'}\subseteq\V{D-W}$ with $\Abs{Z^2_{i,\xi,\xi'}}\leq 8\Fkt{f_P}{t}$ such that every directed $\V{W_0}$-path $P$ in $D-Z^2_{i,\xi,\xi'}$ whose endpoints belong to different tiles of $\Class_i$ contains a vertex of some tile in $\MarkedTiles{2}_{i,\xi,\xi'}$.

Let $W'$ be the largest [def:slice]slice of $W$ such that no tile of $\Class_i$ contains a vertex of $\Perimeter{W'}$.
Let us consider the [def:auxiliarydigraphI]auxiliary digraph $\Fkt{D^2_i}{W'}$ with the set $X_{\text{II}}^i$ of newly added vertices.
By applying <ref> to the set $X_{\text{II}}^i$ in $\Fkt{D^2_i}{W'}$, we either find a set $Z$ of size at most $8\Fkt{f_P}{t}$ that hits all directed $X_{\text{II}}^i$-paths, or there exists a family $\mathcal{J}'$ of $4\Fkt{f_P}{t}$ pairwise disjoint directed $X_{\text{II}}^i$-paths in $\Fkt{D^2_i}{W'}$.

Let us assume the latter.
Then, by construction of $\Fkt{D^2_i}{W'}$, no path in $\mathcal{J}'$ contains a vertex of $W_2$.
Back in the digraph $D$, let us consider the [def:tilingII]tier II tiling $\Tiling''\coloneqq \TierIITiling{\Tiling,i,w}{W'}$ of width $w$ of $W''\coloneqq \InducedSubgraph{W'}{\Tiling,i}$.
Notice that, by choice of $k$, $W''$ still contains a cylindrical $\Fkt{f_W}{t}$-wall $W'''$ such that the perimeter of every tile $T\in\Class_i$, for which $x_T$ is an endpoint of some path in $\mathcal{J}'$, bounds a cell of $W'''$.
Let $\Tiling''$ be a tiling of $W'''$ such that the perimeter of every $T\in\Class_i$, for which $x_T$ is an endpoint of a path in $\mathcal{J}'$, is the centre of some tile in $\Tiling''$.
We now consider a four colouring $\Set{\Class_1',\dots,\Class_4'}$ of $\Tiling''$.
Then there must exist $j\in[1,4]$ and a family $\mathcal{J}''$ of size $\Fkt{f_P}{t}$ such that the starting points of every path in $\mathcal{J}''$ belongs to a tile of $\Class_i$ whose perimeter is the centre of a tile in $\Class_j'$.
For every $J''\in\mathcal{J}''$ let $T_1,T_2\in\Class_i$ be the two tiles such that $J'$ is a directed $x_{T_1}$-$x_{T_2}$-path.
We can now find a directed path $J$ that starts on the perimeter of of $T_1$, ends on the perimeter of $T_2$, and is internally disjoint from $W'''$.
Hence we find a family $\mathcal{J}$ of pairwise disjoint directed $W'''$-paths whose endpoints all lie on the centres of distinct tiles of $\Tiling''$ and that all start at the centres of tiles from $\Class_j'$.
So we may apply <ref> to find a $K_{t,t}$-[def:matchingminor]matching minor grasped by [def:split]$\Split{W'''}$.
Moreover, with $W'''\subseteq W\subseteq W_0$, we have found a $K_{t,t}$-matching minor grasped by $\Split{W_0}$.

Therefore we may assume that we find a set $Z$ of size at most $8\Fkt{f_P}{t}$ that hits all directed $X_{\text{II}}^i$-paths.
Let $Z^2_{i,\xi,\xi'}\coloneqq Z\cap \V{D}$, and $\MarkedTiles{2}_{i,\xi,\xi'}\coloneqq\CondSet{T\in\Tiling}{x_T\in Z}$.
Since $\Abs{Z}\leq 2\Fkt{f_P}{t}$, the bounds on the two sets follow immediately.
Moreover, since $Z$ meets every directed $X_{\text{II}}^i$-path in $D'$, every directed path with endpoints in distinct tiles of $\Class_i$ which is otherwise disjoint from $W_0$ must contain a vertex from $Z^2_{i,\xi,\xi'}$ or meet a tile from $\MarkedTiles{2}_{i,\xi,\xi'}$.

§ STEP 2: CROSSES AND BIPARTITE $K_{3,3}$-FREE GRAPHS

In the previous section, we have taken care of long jumps over our wall.
The proofs presented there are closely related to those necessary for the Directed Flat Wall Theorem.
This section is dedicated to the second step: removing local crosses from our wall.

Non-Pfaffian Cells and Tiles

The two techniques of changing the perfect matching and applying [def:braces]tight cut contractions, are of immediate importance for the first step towards controlling the crossings over our wall.

Let $w\in\N$ be some positive integer.
In the following we will say that a slice $W'$ of some cylindrical wall $W$ in some digraph $D$ is clean if there is no $w$-long jump over $W'$ in $D$.
We say that the slice $W'$ is proper if it does not contain the perimeter of $W$.

A first step is to localise crossings to be able to use them for routing.
Ideally, we want our crossings to occur `over', or, more precisely, within the `attachment' of a single cell in our wall.
However, this is not necessarily possible, as there might still exist short jumps even over a clean slice.
So instead, we consider crossings over tiles and then use a tier II tiling to force these crossings into a single cell.
For this, we need the following:

* We need a proper definition of an `attachment',

* there needs to be a way to use crossings, or in other words conformal bisubdivisions of $K_{3,3}$ as seen in <ref>, even if these are not $M$-conformal, and

* we need to show that the existence of a short jump with both endpoints in a tile immediately forces the existence of a conformal $K_{3,3}$ subdivision within the attachment of the tile.

It is important to note that, since we are interested in conformal bisubdivisions of $K_{3,3}$, it suffices to work on a brace.
We require some additional observations on matching walls in bipartite graphs with perfect matchings and their braces.

Let $k\in\N$ be positive integer with $k\geq 2$, $W$ be a [def:matchingwall]matching $k+2$-wall with canonical matching $M$, $G_k$ be a subdivision the cylindrical grid of order $k$, and $W'$ be the unique proper [def:slice]slice of width $k$ of $W$.
Then the following statements are true:

* $W$ is an expansion of a brace $J$,

* $J$ contains a remnant $H$ of $W'$, and

* $H$ is a cubic graph that is an expansion of $\Split{G_k}$.

We start by proving that $W$ has a unique brace that is not isomorphic to $C_4$.
To do this, let $X\subseteq\V{W}$ be any set that induces a non-trivial tight cut in $W$.
Moreover, let us assume that $\Complement{X}$ contains more degree-$3$-vertices of $W$ than $X$, while, in case both shores contain the same number of such vertices, let $X$ be chosen arbitrarily.
Note that $X$ cannot contain vertices from two distinct vertical cycles of $W$, since we may switch their matchings independently and thus could always force at least two matching edges to lie in $\CutG{W}{X}$.
Similarly, for every pair $P_1$, $P_2$ of horizontal paths, we can always find an $M$-conformal cycle $C$ in $W$ that contains both $P_1$ and $P_2$.
Hence $X$ cannot contain vertices from both $P_1$ and $P_2$, as otherwise it would either contain vertices from two distinct vertical cycles, or we could switch the matching along $C$ to force at least two edges of the new matching into $\CutG{W}{X}$.
Hence $\InducedSubgraph{W}{X}$ must be one of three things:

* an induced subpath of some vertical cycle of $W$,

* an induced subpath of some horizontal path of $W$, or

* a subdivided star that contains exactly one degree-$3$-vertex $s$ of $W$ and $s$ lies at the centre of the star.

In the first two cases, contracting $\Complement{X}$ clearly results in a cycle of even length and thus all of its braces are isomorphic to $C_4$.
In the last case observe that, in order for $X$ to induce a [def:braces]tight cut, the leaf vertices of the star must all have the same colour by <ref> and thus, since we are forced to have an imbalance of exactly one, the centre of the star must be part of the [obs:tightcutminoritymajority]majority as well.

(a1) [v:main] ;

(b1) [v:mainempty,position=45:13mm from a1] ;
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(b3) [v:mainempty,position=315:13mm from a1] ;

(b4) [v:mainempty,position=0:33mm from a1] ;

(a2) [v:main,position=135:13mm from b4] ;
(a3) [v:main,position=180:13mm from b4] ;
(a4) [v:main,position=225:13mm from b4] ;

[e:coloredborder] (a1) to (b1);
[e:coloredborder] (a2) to (b4);
[e:coloredborder] (a3) to (b2);
[e:coloredborder] (a4) to (b3);

[e:colored,color=BostonUniversityRed] (a1) to (b1);
[e:colored,color=BostonUniversityRed] (a2) to (b4);
[e:colored,color=BostonUniversityRed] (a3) to (b2);
[e:colored,color=BostonUniversityRed] (a4) to (b3);

[e:main] (a1) to (b2);
[e:main] (a1) to (b3);
[e:main] (a3) to (b4);
[e:main] (a4) to (b4);
[e:main] (a2) to (b1);

The tight cut contraction of a star.

So when contracting $\Complement{X}$, this case yields a bisubdivision of the graph in <ref>.
It is straightforward to see that every brace of this graph must be isomorphic to $C_4$.
As we have seen, one of the two tight cut contractions of any chosen non-trivial tight cut in $W$ yields exclusively braces which are isomorphic to $C_4$.
We have also seen that no such shore can contain more then one degree-$3$-vertex of $W$, and thus the other shore must still contain a remnant of some brace $J$ which contains all other degree-$3$-vertices.
Hence $W$ contains a brace $J$ which has at least as many degree-$3$-vertices as $W$ has, and this brace $J$ must be unique.
Our claim now follows from the uniqueness of the tight cut decomposition [22].

Next, observe that there are exactly two vertical cycles $C_1$ and $C_2$ in $W$ which contain degree-$3$-vertices that are linked by paths of even length along $C_i$, whose internal vertices are all of degree two in $W$.
These two cycles are in fact exactly those which form the perimeter of $W$.
Hence the second and third point from the assertion follow immediately.

Let $B$ be a bipartite graph with a perfect matching $M$, $D\coloneqq \DirM{B}{M}$, and $W$ be a proper slice of some cylindrical wall $W'$ in $D$.
Let $B'$ be a host of $\Split{W'}$ in $B$, and let $\widetilde{W}$ be the remnant of $\Split{W}$ in $B'$.

Now let $H\subseteq B$ and $M'\in\Perf{B}$ be chosen such that

* $\Split{W}$ is $M'$-conformal in $B$,

* $\E{B-\Split{W'}}\cap M'\subseteq M$, and

* $H$ is an induced $M'$-conformal subgraph of $\Split{W'}$ such that its outer face, i.e. the face of $H$ that contains $\Split{W'}-H$ in the canonical embedding of $\Split{W}$, is an $M'$-conformal cycle.

Let $H'$ be the remnant of $H$ in $B'$.
The attachment of $H$ over $W$ in $B$, denoted by $\Attachment{B,\Split{W}}{H}$, is the elementary component of $B'-\Brace{\widetilde{W}-H'}$ that contains the outer face of $H'$.

Let $B$ be a planar brace and $C$ a conformal and separating cycle in $B$, then $\Inner{B}{C}$ and $\Outer{B}{C}$ are matching covered.

Let $W$ be a [def:matchingwall]matching wall and $H$ be an induced $M'$-conformal subgraph of $W$ whose outer face is an $M$-conformal cycle $C$ where $M'\in\Perf{W}$.
Then $C$ is conformal and separating in $W$ and thus, by <ref>, $H$ is matching covered.
That means, for every $H\subseteq W$ for which an attachment exists, we know that $H'$, as in the definition above, belongs to this attachment.

Since we are interested in tilings and their centres especially, we need to show that the attachment of a cell and the attachment of a tile are well defined objects.
Moreover, we would like to know that the attachment of a tile contains the attachment of its centre.
To this end, we introduce the following two lemmas.

Let $h,k,w\in\N$ be positive integers and $W$ be a proper [def:slice]slice of width $k$ of some [def:matchingwall]matching $h$-wall where $h\geq k+2$.
Let $M$ be the canonical matching of $W$ and [def:tiling]$\Tiling=\Tiling_{W,k,w,\xi,\xi'}$ for $\xi,\xi'\in[1,w+1]$.
At last let $T\in\Tiling$ be any [def:tile]tile that is completely contained in $W$ and let $Q$ be the vertical cycle of $W$ that contains the leftmost part of $\Perimeter{T}$.
Then $M_T\coloneqq M\Delta \E{Q}$ is a perfect matching of $W$ such that $\Perimeter{T}$ contains an $M_T$-conformal cycle $C_T$ and $T-\Brace{\Perimeter{T}-C_T}$ is an $M_T$-conformal matching covered subgraph of $W$ whose outer face is bounded by $C_T$.

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(s33) [v:main,position=0:6.5mm from v33,color=BostonUniversityRed,opacity=0.35] ;
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(v315) [v:main,position=0:13mm from v314,color=Gray] ;
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(v10-7) [v:main,position=0:13mm from v10-6] ;
(v10-8) [v:main,position=0:13mm from v10-7] ;
(v10-9) [v:main,position=0:13mm from v10-8] ;
(v10-10) [v:main,position=0:13mm from v10-9] ;
(v10-11) [v:main,position=0:13mm from v10-10] ;
(v10-12) [v:main,position=0:13mm from v10-11] ;
(v10-13) [v:main,position=0:13mm from v10-12,color=BostonUniversityRed] ;
(v10-14) [v:main,position=0:13mm from v10-13,color=BostonUniversityRed] ;
(v10-15) [v:main,position=0:13mm from v10-14,color=Gray] ;
(v10-16) [v:main,position=0:13mm from v10-15,color=Gray] ;

(v11-1) [v:main,position=270:8mm from v10-1,color=Gray] ;
(v11-2) [v:main,position=0:13mm from v11-1,color=Gray] ;
(v11-3) [v:ghost,position=0:13mm from v11-2,color=BostonUniversityRed] ;
(v11-4) [v:ghost,position=0:13mm from v11-3,color=BostonUniversityRed] ;
(s11-3) [v:main,position=0:6.5mm from v11-3,color=BostonUniversityRed] ;
(s11-4) [v:main,position=270:4mm from v11-4,color=BostonUniversityRed] ;
(v11-5) [v:main,position=0:13mm from v11-4] ;
(v11-6) [v:main,position=0:13mm from v11-5] ;
(v11-7) [v:main,position=0:13mm from v11-6] ;
(v11-8) [v:main,position=0:13mm from v11-7] ;
(v11-9) [v:main,position=0:13mm from v11-8] ;
(v11-10) [v:main,position=0:13mm from v11-9] ;
(v11-11) [v:main,position=0:13mm from v11-10] ;
(v11-12) [v:main,position=0:13mm from v11-11] ;
(v11-13) [v:main,position=0:13mm from v11-12,color=BostonUniversityRed] ;
(v11-14) [v:main,position=0:13mm from v11-13,color=BostonUniversityRed] ;
(v11-15) [v:main,position=0:13mm from v11-14,color=Gray] ;
(v11-16) [v:main,position=0:13mm from v11-15,color=Gray] ;

(v12-1) [v:main,position=270:8mm from v11-1,color=Gray] ;
(v12-2) [v:main,position=0:13mm from v12-1,color=Gray] ;
(v12-3) [v:ghost,position=0:13mm from v12-2,color=BostonUniversityRed] ;
(v12-4) [v:ghost,position=0:13mm from v12-3,color=BostonUniversityRed] ;
(s12-3) [v:main,position=270:4mm from v12-3,color=BostonUniversityRed] ;
(s12-4) [v:main,position=180:6.5mm from v12-4,color=BostonUniversityRed] ;
(v12-5) [v:main,position=0:13mm from v12-4] ;
(v12-6) [v:main,position=0:13mm from v12-5] ;
(v12-7) [v:main,position=0:13mm from v12-6] ;
(v12-8) [v:main,position=0:13mm from v12-7] ;
(v12-9) [v:main,position=0:13mm from v12-8] ;
(v12-10) [v:main,position=0:13mm from v12-9] ;
(v12-11) [v:main,position=0:13mm from v12-10] ;
(v12-12) [v:main,position=0:13mm from v12-11] ;
(v12-13) [v:main,position=0:13mm from v12-12,color=BostonUniversityRed] ;
(v12-14) [v:main,position=0:13mm from v12-13,color=BostonUniversityRed] ;
(v12-15) [v:main,position=0:13mm from v12-14,color=Gray] ;
(v12-16) [v:main,position=0:13mm from v12-15,color=Gray] ;

(v13-1) [v:main,position=270:8mm from v12-1,color=Gray] ;
(v13-2) [v:main,position=0:13mm from v13-1,color=Gray] ;
(v13-3) [v:ghost,position=0:13mm from v13-2,color=BostonUniversityRed] ;
(v13-4) [v:ghost,position=0:13mm from v13-3,color=BostonUniversityRed] ;
(s13-3) [v:main,position=0:6.5mm from v13-3,color=BostonUniversityRed] ;
(s13-4) [v:main,position=270:4mm from v13-4,color=BostonUniversityRed] ;
(v13-5) [v:main,position=0:13mm from v13-4] ;
(v13-6) [v:main,position=0:13mm from v13-5] ;
(v13-7) [v:main,position=0:13mm from v13-6] ;
(v13-8) [v:main,position=0:13mm from v13-7] ;
(v13-9) [v:main,position=0:13mm from v13-8] ;
(v13-10) [v:main,position=0:13mm from v13-9] ;
(v13-11) [v:main,position=0:13mm from v13-10] ;
(v13-12) [v:main,position=0:13mm from v13-11] ;
(v13-13) [v:main,position=0:13mm from v13-12,color=BostonUniversityRed] ;
(v13-14) [v:main,position=0:13mm from v13-13,color=BostonUniversityRed] ;
(v13-15) [v:main,position=0:13mm from v13-14,color=Gray] ;
(v13-16) [v:main,position=0:13mm from v13-15,color=Gray] ;

(v14-1) [v:main,position=270:8mm from v13-1,color=Gray] ;
(v14-2) [v:main,position=0:13mm from v14-1,color=Gray] ;
(v14-3) [v:ghost,position=0:13mm from v14-2,color=BostonUniversityRed] ;
(v14-4) [v:ghost,position=0:13mm from v14-3,color=BostonUniversityRed] ;
(s14-3) [v:main,position=270:4mm from v14-3,color=BostonUniversityRed,opacity=0.35] ;
(s14-4) [v:main,position=180:6.5mm from v14-4,color=BostonUniversityRed,opacity=0.35] ;
(v14-5) [v:main,position=0:13mm from v14-4,color=BostonUniversityRed] ;
(v14-6) [v:main,position=0:13mm from v14-5,color=BostonUniversityRed] ;
(v14-7) [v:main,position=0:13mm from v14-6,color=BostonUniversityRed] ;
(v14-8) [v:main,position=0:13mm from v14-7,color=BostonUniversityRed] ;
(v14-9) [v:main,position=0:13mm from v14-8,color=BostonUniversityRed] ;
(v14-10) [v:main,position=0:13mm from v14-9,color=BostonUniversityRed] ;
(v14-11) [v:main,position=0:13mm from v14-10,color=BostonUniversityRed] ;
(v14-12) [v:main,position=0:13mm from v14-11,color=BostonUniversityRed] ;
(v14-13) [v:main,position=0:13mm from v14-12,color=BostonUniversityRed] ;
(v14-14) [v:main,position=0:13mm from v14-13,color=BostonUniversityRed] ;
(v14-15) [v:main,position=0:13mm from v14-14,color=Gray] ;
(v14-16) [v:main,position=0:13mm from v14-15,color=Gray] ;

(v15-1) [v:main,position=270:8mm from v14-1,color=Gray] ;
(v15-2) [v:main,position=0:13mm from v15-1,color=Gray] ;
(v15-3) [v:ghost,position=0:13mm from v15-2,color=Gray] ;
(v15-4) [v:ghost,position=0:13mm from v15-3,color=Gray] ;
(s15-3) [v:main,position=0:6.5mm from v15-3,color=Gray] ;
(s15-4) [v:main,position=270:4mm from v15-4,color=Gray] ;
(v15-5) [v:main,position=0:13mm from v15-4,color=Gray] ;
(v15-6) [v:main,position=0:13mm from v15-5,color=Gray] ;
(v15-7) [v:main,position=0:13mm from v15-6,color=Gray] ;
(v15-8) [v:main,position=0:13mm from v15-7,color=Gray] ;
(v15-9) [v:main,position=0:13mm from v15-8,color=Gray] ;
(v15-10) [v:main,position=0:13mm from v15-9,color=Gray] ;
(v15-11) [v:main,position=0:13mm from v15-10,color=Gray] ;
(v15-12) [v:main,position=0:13mm from v15-11,color=Gray] ;
(v15-13) [v:main,position=0:13mm from v15-12,color=Gray] ;
(v15-14) [v:main,position=0:13mm from v15-13,color=Gray] ;
(v15-15) [v:main,position=0:13mm from v15-14,color=Gray] ;
(v15-16) [v:main,position=0:13mm from v15-15,color=Gray] ;

(v16-1) [v:main,position=270:8mm from v15-1,color=Gray] ;
(v16-2) [v:main,position=0:13mm from v16-1,color=Gray] ;
(v16-3) [v:ghost,position=0:13mm from v16-2,color=Gray] ;
(v16-4) [v:ghost,position=0:13mm from v16-3,color=Gray] ;
(s16-3) [v:main,position=270:4mm from v16-3,color=Gray] ;
(s16-4) [v:main,position=180:6.5mm from v16-4,color=Gray] ;
(v16-5) [v:main,position=0:13mm from v16-4,color=Gray] ;
(v16-6) [v:main,position=0:13mm from v16-5,color=Gray] ;
(v16-7) [v:main,position=0:13mm from v16-6,color=Gray] ;
(v16-8) [v:main,position=0:13mm from v16-7,color=Gray] ;
(v16-9) [v:main,position=0:13mm from v16-8,color=Gray] ;
(v16-10) [v:main,position=0:13mm from v16-9,color=Gray] ;
(v16-11) [v:main,position=0:13mm from v16-10,color=Gray] ;
(v16-12) [v:main,position=0:13mm from v16-11,color=Gray] ;
(v16-13) [v:main,position=0:13mm from v16-12,color=Gray] ;
(v16-14) [v:main,position=0:13mm from v16-13,color=Gray] ;
(v16-15) [v:main,position=0:13mm from v16-14,color=Gray] ;
(v16-16) [v:main,position=0:13mm from v16-15,color=Gray] ;

(v17-1) [v:ghost,position=225:8mm from v16-1] ;
(v17-3) [v:ghost,position=225:8mm from s16-3] ;
(vv17-3) [v:ghost,position=270:8mm from v16-2] ;
(v17-5) [v:ghost,position=225:8mm from v16-5] ;
(v17-7) [v:ghost,position=225:8mm from v16-7] ;
(v17-9) [v:ghost,position=225:8mm from v16-9] ;
(v17-11) [v:ghost,position=225:8mm from v16-11] ;
(v17-13) [v:ghost,position=225:8mm from v16-13] ;
(v17-15) [v:ghost,position=225:8mm from v16-15] ;

(centre) [fill=BritishRacingGreen,minimum width=16.8mm,minimum height=5.55mm,opacity=0.3,position=270:4mm from v88] ;

[e:main,->,color=Gray] (v11) to (v12);
[e:main,->,color=Gray] (v12) to (vv03);
[e:main,->,color=Gray] (vv17-3) to (s16-3);
[e:main,<-,color=Gray] (s13) to (s14);
[e:main,->,color=Gray] (s14) to (v15);
[e:main,->,color=Gray] (v15) to (v16);
[e:main,->,color=Gray] (v16) to (v17);
[e:main,->,color=Gray] (v17) to (v18);
[e:main,->,color=Gray] (v18) to (v19);
[e:main,->,color=Gray] (v19) to (v110);
[e:main,->,color=Gray] (v110) to (v111);
[e:main,->,color=Gray] (v111) to (v112);
[e:main,->,color=Gray] (v112) to (v113);
[e:main,->,color=Gray] (v113) to (v114);
[e:main,->,color=Gray] (v114) to (v115);
[e:main,->,color=Gray] (v115) to (v116);

[e:main,<-,color=Gray] (v21) to (v22);
[e:main,<-,color=Gray] (v22) to (s23);
[e:main,->,color=Gray] (s23) to (s24);
[e:main,<-,color=Gray] (s14) to (v25);
[e:main,<-,color=Gray] (v25) to (v26);
[e:main,<-,color=Gray] (v26) to (v27);
[e:main,<-,color=Gray] (v27) to (v28);
[e:main,<-,color=Gray] (v28) to (v29);
[e:main,<-,color=Gray] (v29) to (v210);
[e:main,<-,color=Gray] (v210) to (v211);
[e:main,<-,color=Gray] (v211) to (v212);
[e:main,<-,color=Gray] (v212) to (v213);
[e:main,<-,color=Gray] (v213) to (v214);
[e:main,<-,color=Gray] (v214) to (v215);
[e:main,<-,color=Gray] (v215) to (v216);

[e:main,->,color=Gray] (v31) to (v32);
[e:main,->,color=Gray] (v32) to (s23);
[e:main,<-,color=BostonUniversityRed,opacity=0.35] (s33) to (s34);
[e:main,->,color=BostonUniversityRed] (s34) to (v35);
[e:main,->,color=BostonUniversityRed] (v35) to (v36);
[e:main,->,color=BostonUniversityRed] (v36) to (v37);
[e:main,->,color=BostonUniversityRed] (v37) to (v38);
[e:main,->,color=BostonUniversityRed] (v38) to (v39);
[e:main,->,color=BostonUniversityRed] (v39) to (v310);
[e:main,->,color=BostonUniversityRed] (v310) to (v311);
[e:main,->,color=BostonUniversityRed] (v311) to (v312);
[e:main,->,color=BostonUniversityRed] (v312) to (v313);
[e:main,->,color=BostonUniversityRed] (v313) to (v314);
[e:main,->,color=Gray] (v314) to (v315);
[e:main,->,color=Gray] (v315) to (v316);

[e:main,<-,color=Gray] (v41) to (v42);
[e:main,<-,color=Gray] (v42) to (s43);
[e:main,->,color=BostonUniversityRed] (s43) to (s44);
[e:main,<-] (s34) to (v45);
[e:main,<-] (v45) to (v46);
[e:main,<-] (v46) to (v47);
[e:main,<-] (v47) to (v48);
[e:main,<-] (v48) to (v49);
[e:main,<-] (v49) to (v410);
[e:main,<-] (v410) to (v411);
[e:main,<-] (v411) to (v412);
[e:main,<-] (v412) to (v413);
[e:main,<-,color=BostonUniversityRed] (v413) to (v414);
[e:main,<-,color=Gray] (v414) to (v415);
[e:main,<-,color=Gray] (v415) to (v416);

[e:main,->,color=Gray] (v51) to (v52);
[e:main,->,color=Gray] (v52) to (s43);
[e:main,<-,color=BostonUniversityRed] (s53) to (s54);
[e:main,->] (s54) to (v55);
[e:main,->] (v55) to (v56);
[e:main,->] (v56) to (v57);
[e:main,->] (v57) to (v58);
[e:main,->] (v58) to (v59);
[e:main,->] (v59) to (v510);
[e:main,->] (v510) to (v511);
[e:main,->] (v511) to (v512);
[e:main,->] (v512) to (v513);
[e:main,->,color=BostonUniversityRed] (v513) to (v514);
[e:main,->,color=Gray] (v514) to (v515);
[e:main,->,color=Gray] (v515) to (v516);

[e:main,<-,color=Gray] (v61) to (v62);
[e:main,<-,color=Gray] (v62) to (s63);
[e:main,->,color=BostonUniversityRed] (s63) to (s64);
[e:main,<-] (s54) to (v65);
[e:main,<-] (v65) to (v66);
[e:main,<-] (v66) to (v67);
[e:main,<-] (v67) to (v68);
[e:main,<-] (v68) to (v69);
[e:main,<-] (v69) to (v610);
[e:main,<-] (v610) to (v611);
[e:main,<-] (v611) to (v612);
[e:main,<-] (v612) to (v613);
[e:main,<-,color=BostonUniversityRed] (v613) to (v614);
[e:main,<-,color=Gray] (v614) to (v615);
[e:main,<-,color=Gray] (v615) to (v616);

[e:main,->,color=Gray] (v71) to (v72);
[e:main,->,color=Gray] (v72) to (s63);
[e:main,<-,color=BostonUniversityRed] (s73) to (s74);
[e:main,->] (s74) to (v75);
[e:main,->] (v75) to (v76);
[e:main,->] (v76) to (v77);
[e:main,->] (v77) to (v78);
[e:main,->] (v78) to (v79);
[e:main,->] (v79) to (v710);
[e:main,->] (v710) to (v711);
[e:main,->] (v711) to (v712);
[e:main,->] (v712) to (v713);
[e:main,->,color=BostonUniversityRed] (v713) to (v714);
[e:main,->,color=Gray] (v714) to (v715);
[e:main,->,color=Gray] (v715) to (v716);

[e:main,<-,color=Gray] (v81) to (v82);
[e:main,<-,color=Gray] (v82) to (s83);
[e:main,->,color=BostonUniversityRed] (s83) to (s84);
[e:main,<-] (s74) to (v85);
[e:main,<-] (v85) to (v86);
[e:main,<-] (v86) to (v87);
[e:main,<-,color=BritishRacingGreen] (v87) to (v88);
[e:main,<-,color=BritishRacingGreen] (v88) to (v89);
[e:main,<-] (v89) to (v810);
[e:main,<-] (v810) to (v811);
[e:main,<-] (v811) to (v812);
[e:main,<-] (v812) to (v813);
[e:main,<-,color=BostonUniversityRed] (v813) to (v814);
[e:main,<-,color=Gray] (v814) to (v815);
[e:main,<-,color=Gray] (v815) to (v816);

[e:main,->,color=Gray] (v91) to (v92);
[e:main,->,color=Gray] (v92) to (s83);
[e:main,<-,color=BostonUniversityRed] (s93) to (s94);
[e:main,->] (s94) to (v95);
[e:main,->] (v95) to (v96);
[e:main,->] (v96) to (v97);
[e:main,->,color=BritishRacingGreen] (v97) to (v98);
[e:main,->,color=BritishRacingGreen] (v98) to (v99);
[e:main,->] (v99) to (v910);
[e:main,->] (v910) to (v911);
[e:main,->] (v911) to (v912);
[e:main,->] (v912) to (v913);
[e:main,->,color=BostonUniversityRed] (v913) to (v914);
[e:main,->,color=Gray] (v914) to (v915);
[e:main,->,color=Gray] (v915) to (v916);

[e:main,<-,color=Gray] (v10-1) to (v10-2);
[e:main,<-,color=Gray] (v10-2) to (s10-3);
[e:main,->,color=BostonUniversityRed] (s10-3) to (s10-4);
[e:main,<-] (s94) to (v10-5);
[e:main,<-] (v10-5) to (v10-6);
[e:main,<-] (v10-6) to (v10-7);
[e:main,<-] (v10-7) to (v10-8);
[e:main,<-] (v10-8) to (v10-9);
[e:main,<-] (v10-9) to (v10-10);
[e:main,<-] (v10-10) to (v10-11);
[e:main,<-] (v10-11) to (v10-12);
[e:main,<-] (v10-12) to (v10-13);
[e:main,<-,color=BostonUniversityRed] (v10-13) to (v10-14);
[e:main,<-,color=Gray] (v10-14) to (v10-15);
[e:main,<-,color=Gray] (v10-15) to (v10-16);

[e:main,->,color=Gray] (v11-1) to (v11-2);
[e:main,->,color=Gray] (v11-2) to (s10-3);
[e:main,<-,color=BostonUniversityRed] (s11-3) to (s11-4);
[e:main,->] (s11-4) to (v11-5);
[e:main,->] (v11-5) to (v11-6);
[e:main,->] (v11-6) to (v11-7);
[e:main,->] (v11-7) to (v11-8);
[e:main,->] (v11-8) to (v11-9);
[e:main,->] (v11-9) to (v11-10);
[e:main,->] (v11-10) to (v11-11);
[e:main,->] (v11-11) to (v11-12);
[e:main,->] (v11-12) to (v11-13);
[e:main,->,color=BostonUniversityRed] (v11-13) to (v11-14);
[e:main,->,color=Gray] (v11-14) to (v11-15);
[e:main,->,color=Gray] (v11-15) to (v11-16);

[e:main,<-,color=Gray] (v12-1) to (v12-2);
[e:main,<-,color=Gray] (v12-2) to (s12-3);
[e:main,->,color=BostonUniversityRed] (s12-3) to (s12-4);
[e:main,<-] (s11-4) to (v12-5);
[e:main,<-] (v12-5) to (v12-6);
[e:main,<-] (v12-6) to (v12-7);
[e:main,<-] (v12-7) to (v12-8);
[e:main,<-] (v12-8) to (v12-9);
[e:main,<-] (v12-9) to (v12-10);
[e:main,<-] (v12-10) to (v12-11);
[e:main,<-] (v12-11) to (v12-12);
[e:main,<-] (v12-12) to (v12-13);
[e:main,<-,color=BostonUniversityRed] (v12-13) to (v12-14);
[e:main,<-,color=Gray] (v12-14) to (v12-15);
[e:main,<-,color=Gray] (v12-15) to (v12-16);

[e:main,->,color=Gray] (v13-1) to (v13-2);
[e:main,->,color=Gray] (v13-2) to (s12-3);
[e:main,<-,color=BostonUniversityRed] (s13-3) to (s13-4);
[e:main,->] (s13-4) to (v13-5);
[e:main,->] (v13-5) to (v13-6);
[e:main,->] (v13-6) to (v13-7);
[e:main,->] (v13-7) to (v13-8);
[e:main,->] (v13-8) to (v13-9);
[e:main,->] (v13-9) to (v13-10);
[e:main,->] (v13-10) to (v13-11);
[e:main,->] (v13-11) to (v13-12);
[e:main,->] (v13-12) to (v13-13);
[e:main,->,color=BostonUniversityRed] (v13-13) to (v13-14);
[e:main,->,color=Gray] (v13-14) to (v13-15);
[e:main,->,color=Gray] (v13-15) to (v13-16);

[e:main,<-,color=Gray] (v14-1) to (v14-2);
[e:main,<-,color=Gray] (v14-2) to (s14-3);
[e:main,->,color=BostonUniversityRed,opacity=0.35] (s14-3) to (s14-4);
[e:main,<-,color=BostonUniversityRed] (s13-4) to (v14-5);
[e:main,<-,color=BostonUniversityRed] (v14-5) to (v14-6);
[e:main,<-,color=BostonUniversityRed] (v14-6) to (v14-7);
[e:main,<-,color=BostonUniversityRed] (v14-7) to (v14-8);
[e:main,<-,color=BostonUniversityRed] (v14-8) to (v14-9);
[e:main,<-,color=BostonUniversityRed] (v14-9) to (v14-10);
[e:main,<-,color=BostonUniversityRed] (v14-10) to (v14-11);
[e:main,<-,color=BostonUniversityRed] (v14-11) to (v14-12);
[e:main,<-,color=BostonUniversityRed] (v14-12) to (v14-13);
[e:main,<-,color=BostonUniversityRed] (v14-13) to (v14-14);
[e:main,<-,color=Gray] (v14-14) to (v14-15);
[e:main,<-,color=Gray] (v14-15) to (v14-16);

[e:main,->,color=Gray] (v15-1) to (v15-2);
[e:main,->,color=Gray] (v15-2) to (s14-3);
[e:main,<-,color=Gray] (s15-3) to (s15-4);
[e:main,->,color=Gray] (s15-4) to (v15-5);
[e:main,->,color=Gray] (v15-5) to (v15-6);
[e:main,->,color=Gray] (v15-6) to (v15-7);
[e:main,->,color=Gray] (v15-7) to (v15-8);
[e:main,->,color=Gray] (v15-8) to (v15-9);
[e:main,->,color=Gray] (v15-9) to (v15-10);
[e:main,->,color=Gray] (v15-10) to (v15-11);
[e:main,->,color=Gray] (v15-11) to (v15-12);
[e:main,->,color=Gray] (v15-12) to (v15-13);
[e:main,->,color=Gray] (v15-13) to (v15-14);
[e:main,->,color=Gray] (v15-14) to (v15-15);
[e:main,->,color=Gray] (v15-15) to (v15-16);

[e:main,<-,color=Gray] (v16-1) to (v16-2);
[e:main,<-,color=Gray] (v16-2) to (s16-3);
[e:main,->,color=Gray] (s16-3) to (s16-4);
[e:main,<-,color=Gray] (s15-4) to (v16-5);
[e:main,<-,color=Gray] (v16-5) to (v16-6);
[e:main,<-,color=Gray] (v16-6) to (v16-7);
[e:main,<-,color=Gray] (v16-7) to (v16-8);
[e:main,<-,color=Gray] (v16-8) to (v16-9);
[e:main,<-,color=Gray] (v16-9) to (v16-10);
[e:main,<-,color=Gray] (v16-10) to (v16-11);
[e:main,<-,color=Gray] (v16-11) to (v16-12);
[e:main,<-,color=Gray] (v16-12) to (v16-13);
[e:main,<-,color=Gray] (v16-13) to (v16-14);
[e:main,<-,color=Gray] (v16-14) to (v16-15);
[e:main,<-,color=Gray] (v16-15) to (v16-16);

[e:main,->,color=Gray] (v12) to (v22);
[e:main,<-,color=Gray] (s14) to (s24);
[e:main,->,color=Gray] (v16) to (v26);
[e:main,->,color=Gray] (v18) to (v28);
[e:main,->,color=Gray] (v110) to (v210);
[e:main,->,color=Gray] (v112) to (v212);
[e:main,->,color=Gray] (v114) to (v214);
[e:main,->,color=Gray] (v116) to (v216);

[e:main,->,color=Gray] (v21) to (v31);
[e:main,<-,color=Gray] (s23) to (s33);
[e:main,->,color=Gray] (v25) to (v35);
[e:main,->,color=Gray] (v27) to (v37);
[e:main,->,color=Gray] (v29) to (v39);
[e:main,->,color=Gray] (v211) to (v311);
[e:main,->,color=Gray] (v213) to (v313);
[e:main,->,color=Gray] (v215) to (v315);

[e:main,->,color=Gray] (v32) to (v42);
[e:main,<-,color=BostonUniversityRed] (s34) to (s44);
[e:main,->] (v36) to (v46);
[e:main,->] (v38) to (v48);
[e:main,->] (v310) to (v410);
[e:main,->] (v312) to (v412);
[e:main,->,color=BostonUniversityRed] (v314) to (v414);
[e:main,->,color=Gray] (v316) to (v416);

[e:main,->,color=Gray] (v41) to (v51);
[e:main,<-,color=BostonUniversityRed] (s43) to (s53);
[e:main,->] (v45) to (v55);
[e:main,->] (v47) to (v57);
[e:main,->] (v49) to (v59);
[e:main,->] (v411) to (v511);
[e:main,->,color=BostonUniversityRed] (v413) to (v513);
[e:main,->,color=Gray] (v415) to (v515);

[e:main,->,color=Gray] (v52) to (v62);
[e:main,<-,color=BostonUniversityRed] (s54) to (s64);
[e:main,->] (v56) to (v66);
[e:main,->] (v58) to (v68);
[e:main,->] (v510) to (v610);
[e:main,->] (v512) to (v612);
[e:main,->,color=BostonUniversityRed] (v514) to (v614);
[e:main,->,color=Gray] (v516) to (v616);

[e:main,->,color=Gray] (v61) to (v71);
[e:main,<-,color=BostonUniversityRed] (s63) to (s73);
[e:main,->] (v65) to (v75);
[e:main,->] (v67) to (v77);
[e:main,->] (v69) to (v79);
[e:main,->] (v611) to (v711);
[e:main,->,color=BostonUniversityRed] (v613) to (v713);
[e:main,->,color=Gray] (v615) to (v715);

[e:main,->,color=Gray] (v72) to (v82);
[e:main,<-,color=BostonUniversityRed] (s74) to (s84);
[e:main,->] (v76) to (v86);
[e:main,->] (v78) to (v88);
[e:main,->] (v710) to (v810);
[e:main,->] (v712) to (v812);
[e:main,->,color=BostonUniversityRed] (v714) to (v814);
[e:main,->,color=Gray] (v716) to (v816);

[e:main,->,color=Gray] (v81) to (v91);
[e:main,<-,color=BostonUniversityRed] (s83) to (s93);
[e:main,->] (v85) to (v95);
[e:main,->,color=BritishRacingGreen] (v87) to (v97);
[e:main,->,color=BritishRacingGreen] (v89) to (v99);
[e:main,->] (v811) to (v911);
[e:main,->,color=BostonUniversityRed] (v813) to (v913);
[e:main,->,color=Gray] (v815) to (v915);

[e:main,->,color=Gray] (v92) to (v10-2);
[e:main,<-,color=BostonUniversityRed] (s94) to (s10-4);
[e:main,->] (v96) to (v10-6);
[e:main,->] (v98) to (v10-8);
[e:main,->] (v910) to (v10-10);
[e:main,->] (v912) to (v10-12);
[e:main,->,color=BostonUniversityRed] (v914) to (v10-14);
[e:main,->,color=Gray] (v916) to (v10-16);

[e:main,->,color=Gray] (v10-1) to (v11-1);
[e:main,<-,color=BostonUniversityRed] (s10-3) to (s11-3);
[e:main,->] (v10-5) to (v11-5);
[e:main,->] (v10-7) to (v11-7);
[e:main,->] (v10-9) to (v11-9);
[e:main,->] (v10-11) to (v11-11);
[e:main,->,color=BostonUniversityRed] (v10-13) to (v11-13);
[e:main,->,color=Gray] (v10-15) to (v11-15);

[e:main,->,color=Gray] (v11-2) to (v12-2);
[e:main,<-,color=BostonUniversityRed] (s11-4) to (s12-4);
[e:main,->] (v11-6) to (v12-6);
[e:main,->] (v11-8) to (v12-8);
[e:main,->] (v11-10) to (v12-10);
[e:main,->] (v11-12) to (v12-12);
[e:main,->,color=BostonUniversityRed] (v11-14) to (v12-14);
[e:main,->,color=Gray] (v11-16) to (v12-16);

[e:main,->,color=Gray] (v12-1) to (v13-1);
[e:main,<-,color=BostonUniversityRed] (s12-3) to (s13-3);
[e:main,->] (v12-5) to (v13-5);
[e:main,->] (v12-7) to (v13-7);
[e:main,->] (v12-9) to (v13-9);
[e:main,->] (v12-11) to (v13-11);
[e:main,->,color=BostonUniversityRed] (v12-13) to (v13-13);
[e:main,->,color=Gray] (v12-15) to (v13-15);

[e:main,->,color=Gray] (v13-2) to (v14-2);
[e:main,<-,color=BostonUniversityRed,opacity=0.35] (s13-4) to (s14-4);
[e:main,->] (v13-6) to (v14-6);
[e:main,->] (v13-8) to (v14-8);
[e:main,->] (v13-10) to (v14-10);
[e:main,->] (v13-12) to (v14-12);
[e:main,->,color=BostonUniversityRed] (v13-14) to (v14-14);
[e:main,->,color=Gray] (v13-16) to (v14-16);

[e:main,->,color=Gray] (v14-1) to (v15-1);
[e:main,<-,color=Gray] (s14-3) to (s15-3);
[e:main,->,color=Gray] (v14-5) to (v15-5);
[e:main,->,color=Gray] (v14-7) to (v15-7);
[e:main,->,color=Gray] (v14-9) to (v15-9);
[e:main,->,color=Gray] (v14-11) to (v15-11);
[e:main,->,color=Gray] (v14-13) to (v15-13);
[e:main,->,color=Gray] (v14-15) to (v15-15);

[e:main,->,color=Gray] (v15-2) to (v16-2);
[e:main,<-,color=Gray] (s15-4) to (s16-4);
[e:main,->,color=Gray] (v15-6) to (v16-6);
[e:main,->,color=Gray] (v15-8) to (v16-8);
[e:main,->,color=Gray] (v15-10) to (v16-10);
[e:main,->,color=Gray] (v15-12) to (v16-12);
[e:main,->,color=Gray] (v15-14) to (v16-14);
[e:main,->,color=Gray] (v15-16) to (v16-16);

[e:main,bend left=20,->,color=Gray] (v01) to (v11);
[e:main,bend left=20,<-,color=Gray] (v03) to (s13);
[e:main,bend left=20,->,color=Gray] (v05) to (v15);
[e:main,bend left=20,->,color=Gray] (v07) to (v17);
[e:main,bend left=20,->,color=Gray] (v09) to (v19);
[e:main,bend left=20,->,color=Gray] (v011) to (v111);
[e:main,bend left=20,->,color=Gray] (v013) to (v113);
[e:main,bend left=20,->,color=Gray] (v015) to (v115);

[e:main,bend left=20,->,color=Gray] (v16-1) to (v17-1);
[e:main,bend left=20,<-,color=Gray] (s16-3) to (v17-3);
[e:main,bend left=20,->,color=Gray] (v16-5) to (v17-5);
[e:main,bend left=20,->,color=Gray] (v16-7) to (v17-7);
[e:main,bend left=20,->,color=Gray] (v16-9) to (v17-9);
[e:main,bend left=20,->,color=Gray] (v16-11) to (v17-11);
[e:main,bend left=20,->,color=Gray] (v16-13) to (v17-13);
[e:main,bend left=20,->,color=Gray] (v16-15) to (v17-15);

The tile $T$ in the $M_T$-direction of $W$.

To see that the statement is true, observe that the boundary of $T$ consists of a cycle and two vertices, one of them being the starting vertex $s$ of the subpath of the upper path in $\Perimeter{T}$, and the other one being the endpoint $t$ of the subpath of the lower path in $\Perimeter{T}$.
By definition, $Q$ is the left path of $\Perimeter{T}$ and $L\coloneqq Q\cap\Perimeter{T}$.
Then both $s$ and $t$ belong to $L$.
See <ref> for an illustration.
Now consider $L'\coloneqq L-s-t$ and note that $\Split{L'}$ contains exactly two vertices which do not belong to the unique cycle of $\Split{\Perimeter{T}}$.
Let $P$ be the path obtained from $\Split{L'}$ by removing these two vertices.
Since we removed both endpoints of an $M$-conformal path, where $M$ is the canonical matching of $W$, the result is an internally $M$-conformal path.
Let $C$ be the unique cycle of $\Split{\Perimeter{T}}$, then, as $\Perimeter{T}-Q$ is a directed path, $C-P$ is another internally $M$-conformal path.
Now consider the perfect matching $M_T$.
Then every internally $M$-conformal subpath of $Q$ has now become an $M_T$-conformal path.
So now $C-P$ is internally $M_T$-conformal, while $P$ is $M_T$-conformal.
Hence $C$ is $M_T$-conformal as required.
See <ref> for an illustration.

(LU) [v:ghost]

(v33) [v:ghost,color=Gray] ;
(v34) [v:ghost,position=0:13mm from v33,color=Gray] ;
(s33) [v:ghost,position=0:6.5mm from v33,color=Gray,opacity=0.35] ;
(s34) [v:main,position=270:5mm from v34,color=Gray] ;
(v35) [v:main,position=0:13mm from v34,color=Gray] ;
(v36) [v:main,position=0:13mm from v35,color=Gray] ;
(v37) [v:main,position=0:13mm from v36,color=Gray] ;
(v38) [v:main,position=0:13mm from v37,color=Gray] ;
(v39) [v:main,position=0:13mm from v38,color=Gray] ;
(v310) [v:main,position=0:13mm from v39,color=Gray] ;
(v311) [v:main,position=0:13mm from v310,color=Gray] ;
(v312) [v:main,position=0:13mm from v311,color=Gray] ;
(v313) [v:main,position=0:13mm from v312,color=Gray] ;
(v314) [v:main,position=0:13mm from v313,color=Gray] ;

(v43) [v:ghost,position=270:10mm from v33,color=Gray] ;
(v44) [v:ghost,position=0:13mm from v43,color=Gray] ;
(s43) [v:main,position=270:5mm from v43,color=Gray] ;
(s44) [v:main,position=180:6.5mm from v44,color=Gray] ;
(v45) [v:main,position=0:13mm from v44] ;
(v46) [v:main,position=0:13mm from v45] ;
(v47) [v:main,position=0:13mm from v46] ;
(v48) [v:main,position=0:13mm from v47] ;
(v49) [v:main,position=0:13mm from v48] ;
(v410) [v:main,position=0:13mm from v49] ;
(v411) [v:main,position=0:13mm from v410] ;
(v412) [v:main,position=0:13mm from v411] ;
(v413) [v:main,position=0:13mm from v412,color=Gray] ;
(v414) [v:main,position=0:13mm from v413,color=Gray] ;

(v53) [v:ghost,position=270:10mm from v43,color=Gray] ;
(v54) [v:ghost,position=0:13mm from v53,color=Gray] ;
(s53) [v:main,position=0:6.5mm from v53,color=Gray] ;
(s54) [v:main,position=270:5mm from v54,color=Gray] ;
(v55) [v:main,position=0:13mm from v54] ;
(v56) [v:main,position=0:13mm from v55] ;
(v57) [v:main,position=0:13mm from v56] ;
(v58) [v:main,position=0:13mm from v57] ;
(v59) [v:main,position=0:13mm from v58] ;
(v510) [v:main,position=0:13mm from v59] ;
(v511) [v:main,position=0:13mm from v510] ;
(v512) [v:main,position=0:13mm from v511] ;
(v513) [v:main,position=0:13mm from v512,color=Gray] ;
(v514) [v:main,position=0:13mm from v513,color=Gray] ;

(v63) [v:ghost,position=270:10mm from v53,color=Gray] ;
(v64) [v:ghost,position=0:13mm from v63,color=Gray] ;
(s63) [v:main,position=270:5mm from v63,color=Gray] ;
(s64) [v:main,position=180:6.5mm from v64,color=Gray] ;
(v65) [v:main,position=0:13mm from v64] ;
(v66) [v:main,position=0:13mm from v65] ;
(v67) [v:main,position=0:13mm from v66] ;
(v68) [v:main,position=0:13mm from v67] ;
(v69) [v:main,position=0:13mm from v68] ;
(v610) [v:main,position=0:13mm from v69] ;
(v611) [v:main,position=0:13mm from v610] ;
(v612) [v:main,position=0:13mm from v611] ;
(v613) [v:main,position=0:13mm from v612,color=Gray] ;
(v614) [v:main,position=0:13mm from v613,color=Gray] ;

(v73) [v:ghost,position=270:10mm from v63,color=Gray] ;
(v74) [v:ghost,position=0:13mm from v73,color=Gray] ;
(s73) [v:main,position=0:6.5mm from v73,color=Gray] ;
(s74) [v:main,position=270:5mm from v74,color=Gray] ;
(v75) [v:main,position=0:13mm from v74] ;
(v76) [v:main,position=0:13mm from v75] ;
(v77) [v:main,position=0:13mm from v76] ;
(v78) [v:main,position=0:13mm from v77] ;
(v79) [v:main,position=0:13mm from v78] ;
(v710) [v:main,position=0:13mm from v79] ;
(v711) [v:main,position=0:13mm from v710] ;
(v712) [v:main,position=0:13mm from v711] ;
(v713) [v:main,position=0:13mm from v712,color=Gray] ;
(v714) [v:main,position=0:13mm from v713,color=Gray] ;

(v83) [v:ghost,position=270:10mm from v73,color=Gray] ;
(v84) [v:ghost,position=0:13mm from v83,color=Gray] ;
(s83) [v:main,position=270:5mm from v83,color=Gray] ;
(s84) [v:main,position=180:6.5mm from v84,color=Gray] ;
(v85) [v:main,position=0:13mm from v84] ;
(v86) [v:main,position=0:13mm from v85] ;
(v87) [v:main,position=0:13mm from v86,color=BritishRacingGreen] ;
(v88) [v:main,position=0:13mm from v87,color=BritishRacingGreen] ;
(v89) [v:main,position=0:13mm from v88,color=BritishRacingGreen] ;
(v810) [v:main,position=0:13mm from v89] ;
(v811) [v:main,position=0:13mm from v810] ;
(v812) [v:main,position=0:13mm from v811] ;
(v813) [v:main,position=0:13mm from v812,color=Gray] ;
(v814) [v:main,position=0:13mm from v813,color=Gray] ;

(v93) [v:ghost,position=270:10mm from v83,color=Gray] ;
(v94) [v:ghost,position=0:13mm from v93,color=Gray] ;
(s93) [v:main,position=0:6.5mm from v93,color=Gray] ;
(s94) [v:main,position=270:5mm from v94,color=Gray] ;
(v95) [v:main,position=0:13mm from v94] ;
(v96) [v:main,position=0:13mm from v95] ;
(v97) [v:main,position=0:13mm from v96,color=BritishRacingGreen] ;
(v98) [v:main,position=0:13mm from v97,color=BritishRacingGreen] ;
(v99) [v:main,position=0:13mm from v98,color=BritishRacingGreen] ;
(v910) [v:main,position=0:13mm from v99] ;
(v911) [v:main,position=0:13mm from v910] ;
(v912) [v:main,position=0:13mm from v911] ;
(v913) [v:main,position=0:13mm from v912,color=Gray] ;
(v914) [v:main,position=0:13mm from v913,color=Gray] ;

(v10-3) [v:ghost,position=270:10mm from v93,color=Gray] ;
(v10-4) [v:ghost,position=0:13mm from v10-3,color=Gray] ;
(s10-3) [v:main,position=270:5mm from v10-3,color=Gray] ;
(s10-4) [v:main,position=180:6.5mm from v10-4,color=Gray] ;
(v10-5) [v:main,position=0:13mm from v10-4] ;
(v10-6) [v:main,position=0:13mm from v10-5] ;
(v10-7) [v:main,position=0:13mm from v10-6] ;
(v10-8) [v:main,position=0:13mm from v10-7] ;
(v10-9) [v:main,position=0:13mm from v10-8] ;
(v10-10) [v:main,position=0:13mm from v10-9] ;
(v10-11) [v:main,position=0:13mm from v10-10] ;
(v10-12) [v:main,position=0:13mm from v10-11] ;
(v10-13) [v:main,position=0:13mm from v10-12,color=Gray] ;
(v10-14) [v:main,position=0:13mm from v10-13,color=Gray] ;

(v11-3) [v:ghost,position=270:10mm from v10-3,color=Gray] ;
(v11-4) [v:ghost,position=0:13mm from v11-3,color=Gray] ;
(s11-3) [v:main,position=0:6.5mm from v11-3,color=Gray] ;
(s11-4) [v:main,position=270:5mm from v11-4,color=Gray] ;
(v11-5) [v:main,position=0:13mm from v11-4] ;
(v11-6) [v:main,position=0:13mm from v11-5] ;
(v11-7) [v:main,position=0:13mm from v11-6] ;
(v11-8) [v:main,position=0:13mm from v11-7] ;
(v11-9) [v:main,position=0:13mm from v11-8] ;
(v11-10) [v:main,position=0:13mm from v11-9] ;
(v11-11) [v:main,position=0:13mm from v11-10] ;
(v11-12) [v:main,position=0:13mm from v11-11] ;
(v11-13) [v:main,position=0:13mm from v11-12,color=Gray] ;
(v11-14) [v:main,position=0:13mm from v11-13,color=Gray] ;

(v12-3) [v:ghost,position=270:10mm from v11-3,color=Gray] ;
(v12-4) [v:ghost,position=0:13mm from v12-3,color=Gray] ;
(s12-3) [v:main,position=270:5mm from v12-3,color=Gray] ;
(s12-4) [v:main,position=180:6.5mm from v12-4,color=Gray] ;
(v12-5) [v:main,position=0:13mm from v12-4] ;
(v12-6) [v:main,position=0:13mm from v12-5] ;
(v12-7) [v:main,position=0:13mm from v12-6] ;
(v12-8) [v:main,position=0:13mm from v12-7] ;
(v12-9) [v:main,position=0:13mm from v12-8] ;
(v12-10) [v:main,position=0:13mm from v12-9] ;
(v12-11) [v:main,position=0:13mm from v12-10] ;
(v12-12) [v:main,position=0:13mm from v12-11] ;
(v12-13) [v:main,position=0:13mm from v12-12,color=Gray] ;
(v12-14) [v:main,position=0:13mm from v12-13,color=Gray] ;

(v13-3) [v:ghost,position=270:10mm from v12-3,color=Gray] ;
(v13-4) [v:ghost,position=0:13mm from v13-3,color=Gray] ;
(s13-3) [v:main,position=0:6.5mm from v13-3,color=Gray] ;
(s13-4) [v:main,position=270:5mm from v13-4,color=Gray] ;
(v13-5) [v:main,position=0:13mm from v13-4] ;
(v13-6) [v:main,position=0:13mm from v13-5] ;
(v13-7) [v:main,position=0:13mm from v13-6] ;
(v13-8) [v:main,position=0:13mm from v13-7] ;
(v13-9) [v:main,position=0:13mm from v13-8] ;
(v13-10) [v:main,position=0:13mm from v13-9] ;
(v13-11) [v:main,position=0:13mm from v13-10] ;
(v13-12) [v:main,position=0:13mm from v13-11] ;
(v13-13) [v:main,position=0:13mm from v13-12,color=Gray] ;
(v13-14) [v:main,position=0:13mm from v13-13,color=Gray] ;

(v14-3) [v:ghost,position=270:10mm from v13-3,color=Gray] ;
(v14-4) [v:ghost,position=0:13mm from v14-3,color=Gray] ;
(s14-3) [v:ghost,position=270:5mm from v14-3,color=Gray,opacity=0.35] ;
(s14-4) [v:ghost,position=180:6.5mm from v14-4,color=Gray,opacity=0.35] ;
(v14-5) [v:main,position=0:13mm from v14-4,color=Gray] ;
(v14-6) [v:main,position=0:13mm from v14-5,color=Gray] ;
(v14-7) [v:main,position=0:13mm from v14-6,color=Gray] ;
(v14-8) [v:main,position=0:13mm from v14-7,color=Gray] ;
(v14-9) [v:main,position=0:13mm from v14-8,color=Gray] ;
(v14-10) [v:main,position=0:13mm from v14-9,color=Gray] ;
(v14-11) [v:main,position=0:13mm from v14-10,color=Gray] ;
(v14-12) [v:main,position=0:13mm from v14-11,color=Gray] ;
(v14-13) [v:main,position=0:13mm from v14-12,color=Gray] ;
(v14-14) [v:main,position=0:13mm from v14-13,color=Gray] ;

(centre) [fill=BritishRacingGreen,minimum width=12.5mm,minimum height=5mm,opacity=0.3,position=270:5mm from v88] ;

[e:main,->,color=Gray] (s34) to (v35);
[e:main,->,color=Gray] (v35) to (v36);
[e:main,->,color=Gray] (v36) to (v37);
[e:main,->,color=Gray] (v37) to (v38);
[e:main,->,color=Gray] (v38) to (v39);
[e:main,->,color=Gray] (v39) to (v310);
[e:main,->,color=Gray] (v310) to (v311);
[e:main,->,color=Gray] (v311) to (v312);
[e:main,->,color=Gray] (v312) to (v313);
[e:main,->,color=Gray] (v313) to (v314);

[e:main,->,color=Gray] (s43) to (s44);
[e:main,<-,color=Gray] (s34) to (v45);
[e:main,<-] (v45) to (v46);
[e:main,<-] (v46) to (v47);
[e:main,<-] (v47) to (v48);
[e:main,<-] (v48) to (v49);
[e:main,<-] (v49) to (v410);
[e:main,<-] (v410) to (v411);
[e:main,<-] (v411) to (v412);
[e:main,<-,color=Gray] (v412) to (v413);
[e:main,<-,color=Gray] (v413) to (v414);

[e:main,<-,color=Gray] (s53) to (s54);
[e:main,->,color=Gray] (s54) to (v55);
[e:main,->] (v55) to (v56);
[e:main,->] (v56) to (v57);
[e:main,->] (v57) to (v58);
[e:main,->] (v58) to (v59);
[e:main,->] (v59) to (v510);
[e:main,->] (v510) to (v511);
[e:main,->] (v511) to (v512);
[e:main,->,color=Gray] (v512) to (v513);
[e:main,->,color=Gray] (v513) to (v514);

[e:main,->,color=Gray] (s63) to (s64);
[e:main,<-,color=Gray] (s54) to (v65);
[e:main,<-] (v65) to (v66);
[e:main,<-] (v66) to (v67);
[e:main,<-] (v67) to (v68);
[e:main,<-] (v68) to (v69);
[e:main,<-] (v69) to (v610);
[e:main,<-] (v610) to (v611);
[e:main,<-] (v611) to (v612);
[e:main,<-,color=Gray] (v612) to (v613);
[e:main,<-,color=Gray] (v613) to (v614);

[e:main,<-,color=Gray] (s73) to (s74);
[e:main,->,color=Gray] (s74) to (v75);
[e:main,->] (v75) to (v76);
[e:main,->] (v76) to (v77);
[e:main,->] (v77) to (v78);
[e:main,->] (v78) to (v79);
[e:main,->] (v79) to (v710);
[e:main,->] (v710) to (v711);
[e:main,->] (v711) to (v712);
[e:main,->,color=Gray] (v712) to (v713);
[e:main,->,color=Gray] (v713) to (v714);

[e:main,->,color=Gray] (s83) to (s84);
[e:main,<-,color=Gray] (s74) to (v85);
[e:main,<-] (v85) to (v86);
[e:main,<-] (v86) to (v87);
[e:main,<-,color=BritishRacingGreen] (v87) to (v88);
[e:main,<-,color=BritishRacingGreen] (v88) to (v89);
[e:main,<-] (v89) to (v810);
[e:main,<-] (v810) to (v811);
[e:main,<-] (v811) to (v812);
[e:main,<-,color=Gray] (v812) to (v813);
[e:main,<-,color=Gray] (v813) to (v814);

[e:main,<-,color=Gray] (s93) to (s94);
[e:main,->,color=Gray] (s94) to (v95);
[e:main,->] (v95) to (v96);
[e:main,->] (v96) to (v97);
[e:main,->,color=BritishRacingGreen] (v97) to (v98);
[e:main,->,color=BritishRacingGreen] (v98) to (v99);
[e:main,->] (v99) to (v910);
[e:main,->] (v910) to (v911);
[e:main,->] (v911) to (v912);
[e:main,->,color=Gray] (v912) to (v913);
[e:main,->,color=Gray] (v913) to (v914);

[e:main,->,color=Gray] (s10-3) to (s10-4);
[e:main,<-,color=Gray] (s94) to (v10-5);
[e:main,<-] (v10-5) to (v10-6);
[e:main,<-] (v10-6) to (v10-7);
[e:main,<-] (v10-7) to (v10-8);
[e:main,<-] (v10-8) to (v10-9);
[e:main,<-] (v10-9) to (v10-10);
[e:main,<-] (v10-10) to (v10-11);
[e:main,<-] (v10-11) to (v10-12);
[e:main,<-,color=Gray] (v10-12) to (v10-13);
[e:main,<-,color=Gray] (v10-13) to (v10-14);

[e:main,<-,color=Gray] (s11-3) to (s11-4);
[e:main,->,color=Gray] (s11-4) to (v11-5);
[e:main,->] (v11-5) to (v11-6);
[e:main,->] (v11-6) to (v11-7);
[e:main,->] (v11-7) to (v11-8);
[e:main,->] (v11-8) to (v11-9);
[e:main,->] (v11-9) to (v11-10);
[e:main,->] (v11-10) to (v11-11);
[e:main,->] (v11-11) to (v11-12);
[e:main,->,color=Gray] (v11-12) to (v11-13);
[e:main,->,color=Gray] (v11-13) to (v11-14);

[e:main,->,color=Gray] (s12-3) to (s12-4);
[e:main,<-,color=Gray] (s11-4) to (v12-5);
[e:main,<-] (v12-5) to (v12-6);
[e:main,<-] (v12-6) to (v12-7);
[e:main,<-] (v12-7) to (v12-8);
[e:main,<-] (v12-8) to (v12-9);
[e:main,<-] (v12-9) to (v12-10);
[e:main,<-] (v12-10) to (v12-11);
[e:main,<-] (v12-11) to (v12-12);
[e:main,<-,color=Gray] (v12-12) to (v12-13);
[e:main,<-,color=Gray] (v12-13) to (v12-14);

[e:main,<-,color=Gray] (s13-3) to (s13-4);
[e:main,->,color=Gray] (s13-4) to (v13-5);
[e:main,->] (v13-5) to (v13-6);
[e:main,->] (v13-6) to (v13-7);
[e:main,->] (v13-7) to (v13-8);
[e:main,->] (v13-8) to (v13-9);
[e:main,->] (v13-9) to (v13-10);
[e:main,->] (v13-10) to (v13-11);
[e:main,->] (v13-11) to (v13-12);
[e:main,->,color=Gray] (v13-12) to (v13-13);
[e:main,->,color=Gray] (v13-13) to (v13-14);

[e:main,<-,color=Gray] (s13-4) to (v14-5);
[e:main,<-,color=Gray] (v14-5) to (v14-6);
[e:main,<-,color=Gray] (v14-6) to (v14-7);
[e:main,<-,color=Gray] (v14-7) to (v14-8);
[e:main,<-,color=Gray] (v14-8) to (v14-9);
[e:main,<-,color=Gray] (v14-9) to (v14-10);
[e:main,<-,color=Gray] (v14-10) to (v14-11);
[e:main,<-,color=Gray] (v14-11) to (v14-12);
[e:main,<-,color=Gray] (v14-12) to (v14-13);
[e:main,<-,color=Gray] (v14-13) to (v14-14);

[e:main,<-,color=Gray] (s34) to (s44);
[e:main,->,color=Gray] (v36) to (v46);
[e:main,->,color=Gray] (v38) to (v48);
[e:main,->,color=Gray] (v310) to (v410);
[e:main,->,color=Gray] (v312) to (v412);
[e:main,->,color=Gray] (v314) to (v414);

[e:main,<-,color=Gray] (s43) to (s53);
[e:main,->] (v45) to (v55);
[e:main,->] (v47) to (v57);
[e:main,->] (v49) to (v59);
[e:main,->] (v411) to (v511);
[e:main,->,color=Gray] (v413) to (v513);

[e:main,<-,color=Gray] (s54) to (s64);
[e:main,->] (v56) to (v66);
[e:main,->] (v58) to (v68);
[e:main,->] (v510) to (v610);
[e:main,->] (v512) to (v612);
[e:main,->,color=Gray] (v514) to (v614);

[e:main,<-,color=Gray] (s63) to (s73);
[e:main,->] (v65) to (v75);
[e:main,->] (v67) to (v77);
[e:main,->] (v69) to (v79);
[e:main,->] (v611) to (v711);
[e:main,->,color=Gray] (v613) to (v713);

[e:main,<-,color=Gray] (s74) to (s84);
[e:main,->] (v76) to (v86);
[e:main,->] (v78) to (v88);
[e:main,->] (v710) to (v810);
[e:main,->] (v712) to (v812);
[e:main,->,color=Gray] (v714) to (v814);

[e:main,<-,color=Gray] (s83) to (s93);
[e:main,->] (v85) to (v95);
[e:main,->,color=BritishRacingGreen] (v87) to (v97);
[e:main,->,color=BritishRacingGreen] (v89) to (v99);
[e:main,->] (v811) to (v911);
[e:main,->,color=Gray] (v813) to (v913);

[e:main,<-,color=Gray] (s94) to (s10-4);
[e:main,->] (v96) to (v10-6);
[e:main,->] (v98) to (v10-8);
[e:main,->] (v910) to (v10-10);
[e:main,->] (v912) to (v10-12);
[e:main,->,color=Gray] (v914) to (v10-14);

[e:main,<-,color=Gray] (s10-3) to (s11-3);
[e:main,->] (v10-5) to (v11-5);
[e:main,->] (v10-7) to (v11-7);
[e:main,->] (v10-9) to (v11-9);
[e:main,->] (v10-11) to (v11-11);
[e:main,->,color=Gray] (v10-13) to (v11-13);

[e:main,<-,color=Gray] (s11-4) to (s12-4);
[e:main,->] (v11-6) to (v12-6);
[e:main,->] (v11-8) to (v12-8);
[e:main,->] (v11-10) to (v12-10);
[e:main,->] (v11-12) to (v12-12);
[e:main,->,color=Gray] (v11-14) to (v12-14);

[e:main,<-,color=Gray] (s12-3) to (s13-3);
[e:main,->] (v12-5) to (v13-5);
[e:main,->] (v12-7) to (v13-7);
[e:main,->] (v12-9) to (v13-9);
[e:main,->] (v12-11) to (v13-11);
[e:main,->,color=Gray] (v12-13) to (v13-13);

[e:main,->,color=Gray] (v13-6) to (v14-6);
[e:main,->,color=Gray] (v13-8) to (v14-8);
[e:main,->,color=Gray] (v13-10) to (v14-10);
[e:main,->,color=Gray] (v13-12) to (v14-12);
[e:main,->,color=Gray] (v13-14) to (v14-14);

(RU) [v:ghost,position=0:74mm from LU]

(v33) [v:ghost,color=Gray] ;
(v34) [v:ghost,position=0:15mm from v33,color=Gray] ;
(s33) [v:ghost,position=0:6.5mm from v33,color=Gray,opacity=0.35] ;
(s34) [v:main,position=270:2mm from v34,color=Gray] ;
(t34) [v:mainempty,position=270:5mm from s34,color=Gray,fill=white] ;
(v35) [v:main,position=0:11mm from v34,color=Gray,fill=white] ;
(w35) [v:mainempty,position=0:5mm from v35,color=Gray] ;
(v36) [v:main,position=0:13mm from v35,color=Gray,fill=white] ;
(w36) [v:main,position=0:5mm from v36,color=Gray] ;
(v37) [v:main,position=0:13mm from v36,color=Gray,fill=white] ;
(w37) [v:main,position=0:5mm from v37,color=Gray] ;
(v38) [v:main,position=0:13mm from v37,color=Gray,fill=white] ;
(w38) [v:main,position=0:5mm from v38,color=Gray] ;
(v39) [v:main,position=0:13mm from v38,color=Gray,fill=white] ;
(w39) [v:main,position=0:5mm from v39,color=Gray] ;
(v310) [v:main,position=0:13mm from v39,color=Gray,fill=white] ;
(w310) [v:main,position=0:5mm from v310,color=Gray] ;
(v311) [v:main,position=0:13mm from v310,color=Gray,fill=white] ;
(w311) [v:main,position=0:5mm from v311,color=Gray] ;
(v312) [v:main,position=0:13mm from v311,color=Gray,fill=white] ;
(w312) [v:main,position=0:5mm from v312,color=Gray] ;
(v313) [v:main,position=0:13mm from v312,color=Gray,fill=white] ;
(w313) [v:main,position=0:5mm from v313,color=Gray] ;
(v314) [v:main,position=0:13mm from v313,color=Gray,fill=white] ;
(w314) [v:main,position=0:5mm from v314,color=Gray] ;

(v43) [v:ghost,position=270:10mm from v33,color=Gray] ;
(v44) [v:ghost,position=0:15mm from v43,color=Gray] ;
(s43) [v:main,position=270:2mm from v43,color=Gray] ;
(t43) [v:main,position=270:5mm from s43,color=Gray,fill=white] ;
(s44) [v:main,position=180:9mm from v44,color=Gray,fill=white] ;
(t44) [v:main,position=180:4mm from v44,color=Gray] ;
(v45) [v:main,position=0:11mm from v44] ;
(w45) [v:mainempty,position=0:5mm from v45] ;
(v46) [v:main,position=0:13mm from v45] ;
(w46) [v:mainempty,position=0:5mm from v46] ;
(v47) [v:main,position=0:13mm from v46] ;
(w47) [v:mainempty,position=0:5mm from v47] ;
(v48) [v:main,position=0:13mm from v47] ;
(w48) [v:mainempty,position=0:5mm from v48] ;
(v49) [v:main,position=0:13mm from v48] ;
(w49) [v:mainempty,position=0:5mm from v49] ;
(v410) [v:main,position=0:13mm from v49] ;
(w410) [v:mainempty,position=0:5mm from v410] ;
(v411) [v:main,position=0:13mm from v410] ;
(w411) [v:mainempty,position=0:5mm from v411] ;
(v412) [v:main,position=0:13mm from v411] ;
(w412) [v:mainempty,position=0:5mm from v412] ;
(v413) [v:main,position=0:13mm from v412,color=Gray] ;
(w413) [v:main,position=0:5mm from v413,color=Gray,fill=white] ;
(v414) [v:main,position=0:13mm from v413,color=Gray] ;
(w414) [v:main,position=0:5mm from v414,color=Gray,fill=white] ;

(v53) [v:ghost,position=270:10mm from v43,color=Gray] ;
(v54) [v:ghost,position=0:15mm from v53,color=Gray] ;
(s53) [v:main,position=0:11mm from v53,color=Gray,fill=white] ;
(t53) [v:main,position=0:6mm from v53,color=Gray] ;
(s54) [v:main,position=270:2mm from v54,color=Gray] ;
(t54) [v:main,position=270:5mm from s54,color=Gray,fill=white] ;
(v55) [v:mainempty,position=0:11mm from v54] ;
(w55) [v:main,position=0:5mm from v55] ;
(v56) [v:mainempty,position=0:13mm from v55] ;
(w57) [v:main,position=0:5mm from v56] ;
(v57) [v:mainempty,position=0:13mm from v56] ;
(w58) [v:main,position=0:5mm from v57] ;
(v58) [v:mainempty,position=0:13mm from v57] ;
(w59) [v:main,position=0:5mm from v58] ;
(v59) [v:mainempty,position=0:13mm from v58] ;
(w510) [v:main,position=0:5mm from v59] ;
(v510) [v:mainempty,position=0:13mm from v59] ;
(w511) [v:main,position=0:5mm from v510] ;
(v511) [v:mainempty,position=0:13mm from v510] ;
(w512) [v:main,position=0:5mm from v511] ;
(v512) [v:mainempty,position=0:13mm from v511] ;
(w513) [v:main,position=0:5mm from v512] ;
(v513) [v:mainempty,position=0:13mm from v512,color=Gray,fill=white] ;
(w514) [v:main,position=0:5mm from v513,color=Gray] ;
(v514) [v:mainempty,position=0:13mm from v513,color=Gray,fill=white] ;
(w515) [v:main,position=0:5mm from v514,color=Gray] ;

(v63) [v:ghost,position=270:10mm from v53,color=Gray] ;
(v64) [v:ghost,position=0:15mm from v63,color=Gray] ;
(s63) [v:main,position=270:2mm from v63,color=Gray] ;
(t63) [v:main,position=270:5mm from s63,color=Gray,fill=white] ;
(s64) [v:main,position=180:9mm from v64,color=Gray,fill=white] ;
(t64) [v:main,position=180:4mm from v64,color=Gray] ;
(v65) [v:main,position=0:11mm from v64] ;
(w65) [v:mainempty,position=0:5mm from v65] ;
(v66) [v:main,position=0:13mm from v65] ;
(w66) [v:mainempty,position=0:5mm from v66] ;
(v67) [v:main,position=0:13mm from v66] ;
(w67) [v:mainempty,position=0:5mm from v67] ;
(v68) [v:main,position=0:13mm from v67] ;
(w68) [v:mainempty,position=0:5mm from v68] ;
(v69) [v:main,position=0:13mm from v68] ;
(w69) [v:mainempty,position=0:5mm from v69] ;
(v610) [v:main,position=0:13mm from v69] ;
(w610) [v:mainempty,position=0:5mm from v610] ;
(v611) [v:main,position=0:13mm from v610] ;
(w611) [v:mainempty,position=0:5mm from v611] ;
(v612) [v:main,position=0:13mm from v611] ;
(w612) [v:mainempty,position=0:5mm from v612] ;
(v613) [v:main,position=0:13mm from v612,color=Gray] ;
(w613) [v:main,position=0:5mm from v613,color=Gray,fill=white] ;
(v614) [v:main,position=0:13mm from v613,color=Gray] ;
(w614) [v:main,position=0:5mm from v614,color=Gray,fill=white] ;

(v73) [v:ghost,position=270:10mm from v63,color=Gray] ;
(v74) [v:ghost,position=0:15mm from v73,color=Gray] ;
(s73) [v:main,position=0:11mm from v73,color=Gray,fill=white] ;
(t73) [v:main,position=0:6mm from v73,color=Gray] ;
(s74) [v:main,position=270:2mm from v74,color=Gray] ;
(t74) [v:main,position=270:5mm from s74,color=Gray,fill=white] ;
(v75) [v:mainempty,position=0:11mm from v74] ;
(w75) [v:main,position=0:5mm from v75] ;
(v76) [v:mainempty,position=0:13mm from v75] ;
(w77) [v:main,position=0:5mm from v76] ;
(v77) [v:mainempty,position=0:13mm from v76] ;
(w78) [v:main,position=0:5mm from v77] ;
(v78) [v:mainempty,position=0:13mm from v77] ;
(w79) [v:main,position=0:5mm from v78] ;
(v79) [v:mainempty,position=0:13mm from v78] ;
(w710) [v:main,position=0:5mm from v79] ;
(v710) [v:mainempty,position=0:13mm from v79] ;
(w711) [v:main,position=0:5mm from v710] ;
(v711) [v:mainempty,position=0:13mm from v710] ;
(w712) [v:main,position=0:5mm from v711] ;
(v712) [v:mainempty,position=0:13mm from v711] ;
(w713) [v:main,position=0:5mm from v712] ;
(v713) [v:mainempty,position=0:13mm from v712,color=Gray,fill=white] ;
(w714) [v:main,position=0:5mm from v713,color=Gray] ;
(v714) [v:mainempty,position=0:13mm from v713,color=Gray,fill=white] ;
(w715) [v:main,position=0:5mm from v714,color=Gray] ;

(v83) [v:ghost,position=270:10mm from v73,color=Gray] ;
(v84) [v:ghost,position=0:15mm from v83,color=Gray] ;
(s83) [v:main,position=270:2mm from v83,color=Gray] ;
(t83) [v:main,position=270:5mm from s83,color=Gray,fill=white] ;
(s84) [v:main,position=180:9mm from v84,color=Gray,fill=white] ;
(t84) [v:main,position=180:4mm from v84,color=Gray] ;
(v85) [v:main,position=0:11mm from v84] ;
(w85) [v:mainempty,position=0:5mm from v85] ;
(v86) [v:main,position=0:13mm from v85] ;
(w86) [v:mainempty,position=0:5mm from v86] ;
(v87) [v:main,position=0:13mm from v86,color=BritishRacingGreen] ;
(w87) [v:mainempty,position=0:5mm from v87,color=BritishRacingGreen,fill=white] ;
(v88) [v:main,position=0:13mm from v87,color=BritishRacingGreen] ;
(w88) [v:mainempty,position=0:5mm from v88,color=BritishRacingGreen,fill=white] ;
(v89) [v:main,position=0:13mm from v88,color=BritishRacingGreen] ;
(w89) [v:mainempty,position=0:5mm from v89] ;
(v810) [v:main,position=0:13mm from v89] ;
(w810) [v:mainempty,position=0:5mm from v810] ;
(v811) [v:main,position=0:13mm from v810] ;
(w811) [v:mainempty,position=0:5mm from v811] ;
(v812) [v:main,position=0:13mm from v811] ;
(w812) [v:mainempty,position=0:5mm from v812] ;
(v813) [v:main,position=0:13mm from v812,color=Gray] ;
(w813) [v:main,position=0:5mm from v813,color=Gray,fill=white] ;
(v814) [v:main,position=0:13mm from v813,color=Gray] ;
(w814) [v:main,position=0:5mm from v814,color=Gray,fill=white] ;

(v93) [v:ghost,position=270:10mm from v83,color=Gray] ;
(v94) [v:ghost,position=0:15mm from v93,color=Gray] ;
(s93) [v:main,position=0:11mm from v93,color=Gray,fill=white] ;
(t93) [v:main,position=0:6mm from v93,color=Gray] ;
(s94) [v:main,position=270:2mm from v94,color=Gray] ;
(t94) [v:main,position=270:5mm from s94,color=Gray,fill=white] ;
(v95) [v:mainempty,position=0:11mm from v94] ;
(w95) [v:main,position=0:5mm from v95] ;
(v96) [v:mainempty,position=0:13mm from v95] ;
(w97) [v:main,position=0:5mm from v96] ;
(v97) [v:mainempty,position=0:13mm from v96,color=BritishRacingGreen,fill=white] ;
(w98) [v:main,position=0:5mm from v97,color=BritishRacingGreen] ;
(v98) [v:mainempty,position=0:13mm from v97,color=BritishRacingGreen,fill=white] ;
(w99) [v:main,position=0:5mm from v98,color=BritishRacingGreen] ;
(v99) [v:mainempty,position=0:13mm from v98,color=BritishRacingGreen,fill=white] ;
(w910) [v:main,position=0:5mm from v99] ;
(v910) [v:mainempty,position=0:13mm from v99] ;
(w911) [v:main,position=0:5mm from v910] ;
(v911) [v:mainempty,position=0:13mm from v910] ;
(w912) [v:main,position=0:5mm from v911] ;
(v912) [v:mainempty,position=0:13mm from v911] ;
(w913) [v:main,position=0:5mm from v912] ;
(v913) [v:mainempty,position=0:13mm from v912,color=Gray,fill=white] ;
(w914) [v:main,position=0:5mm from v913,color=Gray] ;
(v914) [v:mainempty,position=0:13mm from v913,color=Gray,fill=white] ;
(w915) [v:main,position=0:5mm from v914,color=Gray] ;

(v10-3) [v:ghost,position=270:10mm from v93,color=Gray] ;
(v10-4) [v:ghost,position=0:15mm from v10-3,color=Gray] ;
(s10-3) [v:main,position=270:2mm from v10-3,color=Gray] ;
(t10-3) [v:main,position=270:5mm from s10-3,color=Gray,fill=white] ;
(s10-4) [v:main,position=180:9mm from v10-4,color=Gray,fill=white] ;
(t10-4) [v:main,position=180:4mm from v10-4,color=Gray] ;
(v10-5) [v:main,position=0:11mm from v10-4] ;
(w10-5) [v:mainempty,position=0:5mm from v10-5] ;
(v10-6) [v:main,position=0:13mm from v10-5] ;
(w10-6) [v:mainempty,position=0:5mm from v10-6] ;
(v10-7) [v:main,position=0:13mm from v10-6] ;
(w10-7) [v:mainempty,position=0:5mm from v10-7] ;
(v10-8) [v:main,position=0:13mm from v10-7] ;
(w10-8) [v:mainempty,position=0:5mm from v10-8] ;
(v10-9) [v:main,position=0:13mm from v10-8] ;
(w10-9) [v:mainempty,position=0:5mm from v10-9] ;
(v10-10) [v:main,position=0:13mm from v10-9] ;
(w10-10) [v:mainempty,position=0:5mm from v10-10] ;
(v10-11) [v:main,position=0:13mm from v10-10] ;
(w10-11) [v:mainempty,position=0:5mm from v10-11] ;
(v10-12) [v:main,position=0:13mm from v10-11] ;
(w10-12) [v:mainempty,position=0:5mm from v10-12] ;
(v10-13) [v:main,position=0:13mm from v10-12,color=Gray] ;
(w10-13) [v:main,position=0:5mm from v10-13,color=Gray,fill=white] ;
(v10-14) [v:main,position=0:13mm from v10-13,color=Gray] ;
(w10-14) [v:main,position=0:5mm from v10-14,color=Gray,fill=white] ;

(v11-3) [v:ghost,position=270:10mm from v10-3,color=Gray] ;
(v11-4) [v:ghost,position=0:15mm from v11-3,color=Gray] ;
(s11-3) [v:main,position=0:11mm from v11-3,color=Gray,fill=white] ;
(t11-3) [v:main,position=0:6mm from v11-3,color=Gray] ;
(s11-4) [v:main,position=270:2mm from v11-4,color=Gray] ;
(t11-4) [v:main,position=270:5mm from s11-4,color=Gray,fill=white] ;
(v11-5) [v:mainempty,position=0:11mm from v11-4] ;
(w11-5) [v:main,position=0:5mm from v11-5] ;
(v11-6) [v:mainempty,position=0:13mm from v11-5] ;
(w11-7) [v:main,position=0:5mm from v11-6] ;
(v11-7) [v:mainempty,position=0:13mm from v11-6] ;
(w11-8) [v:main,position=0:5mm from v11-7] ;
(v11-8) [v:mainempty,position=0:13mm from v11-7] ;
(w11-9) [v:main,position=0:5mm from v11-8] ;
(v11-9) [v:mainempty,position=0:13mm from v11-8] ;
(w11-10) [v:main,position=0:5mm from v11-9] ;
(v11-10) [v:mainempty,position=0:13mm from v11-9] ;
(w11-11) [v:main,position=0:5mm from v11-10] ;
(v11-11) [v:mainempty,position=0:13mm from v11-10] ;
(w11-12) [v:main,position=0:5mm from v11-11] ;
(v11-12) [v:mainempty,position=0:13mm from v11-11] ;
(w11-13) [v:main,position=0:5mm from v11-12] ;
(v11-13) [v:mainempty,position=0:13mm from v11-12,color=Gray,fill=white] ;
(w11-14) [v:main,position=0:5mm from v11-13,color=Gray] ;
(v11-14) [v:mainempty,position=0:13mm from v11-13,color=Gray,fill=white] ;
(w11-15) [v:main,position=0:5mm from v11-14,color=Gray] ;

(v12-3) [v:ghost,position=270:10mm from v11-3,color=Gray] ;
(v12-4) [v:ghost,position=0:15mm from v12-3,color=Gray] ;
(s12-3) [v:main,position=270:2mm from v12-3,color=Gray] ;
(t12-3) [v:main,position=270:5mm from s12-3,color=Gray,fill=white] ;
(s12-4) [v:main,position=180:9mm from v12-4,color=Gray,fill=white] ;
(t12-4) [v:main,position=180:4mm from v12-4,color=Gray] ;
(v12-5) [v:main,position=0:11mm from v12-4] ;
(w12-5) [v:mainempty,position=0:5mm from v12-5] ;
(v12-6) [v:main,position=0:13mm from v12-5] ;
(w12-6) [v:mainempty,position=0:5mm from v12-6] ;
(v12-7) [v:main,position=0:13mm from v12-6] ;
(w12-7) [v:mainempty,position=0:5mm from v12-7] ;
(v12-8) [v:main,position=0:13mm from v12-7] ;
(w12-8) [v:mainempty,position=0:5mm from v12-8] ;
(v12-9) [v:main,position=0:13mm from v12-8] ;
(w12-9) [v:mainempty,position=0:5mm from v12-9] ;
(v12-10) [v:main,position=0:13mm from v12-9] ;
(w12-10) [v:mainempty,position=0:5mm from v12-10] ;
(v12-11) [v:main,position=0:13mm from v12-10] ;
(w12-11) [v:mainempty,position=0:5mm from v12-11] ;
(v12-12) [v:main,position=0:13mm from v12-11] ;
(w12-12) [v:mainempty,position=0:5mm from v12-12] ;
(v12-13) [v:main,position=0:13mm from v12-12,color=Gray] ;
(w12-13) [v:main,position=0:5mm from v12-13,color=Gray,fill=white] ;
(v12-14) [v:main,position=0:13mm from v12-13,color=Gray] ;
(w12-14) [v:main,position=0:5mm from v12-14,color=Gray,fill=white] ;

(v13-3) [v:ghost,position=270:10mm from v12-3,color=Gray] ;
(v13-4) [v:ghost,position=0:15mm from v13-3,color=Gray] ;
(s13-3) [v:main,position=0:11mm from v13-3,color=Gray,fill=white] ;
(t13-3) [v:main,position=0:6mm from v13-3,color=Gray] ;
(s13-4) [v:main,position=270:2mm from v13-4,color=Gray] ;
(t13-4) [v:main,position=270:5mm from s13-4,color=Gray,fill=white] ;
(v13-5) [v:mainempty,position=0:11mm from v13-4] ;
(w13-5) [v:main,position=0:5mm from v13-5] ;
(v13-6) [v:mainempty,position=0:13mm from v13-5] ;
(w13-7) [v:main,position=0:5mm from v13-6] ;
(v13-7) [v:mainempty,position=0:13mm from v13-6] ;
(w13-8) [v:main,position=0:5mm from v13-7] ;
(v13-8) [v:mainempty,position=0:13mm from v13-7] ;
(w13-9) [v:main,position=0:5mm from v13-8] ;
(v13-9) [v:mainempty,position=0:13mm from v13-8] ;
(w13-10) [v:main,position=0:5mm from v13-9] ;
(v13-10) [v:mainempty,position=0:13mm from v13-9] ;
(w13-11) [v:main,position=0:5mm from v13-10] ;
(v13-11) [v:mainempty,position=0:13mm from v13-10] ;
(w13-12) [v:main,position=0:5mm from v13-11] ;
(v13-12) [v:mainempty,position=0:13mm from v13-11] ;
(w13-13) [v:main,position=0:5mm from v13-12] ;
(v13-13) [v:mainempty,position=0:13mm from v13-12,color=Gray,fill=white] ;
(w13-14) [v:main,position=0:5mm from v13-13,color=Gray] ;
(v13-14) [v:mainempty,position=0:13mm from v13-13,color=Gray,fill=white] ;
(w13-15) [v:main,position=0:5mm from v13-14,color=Gray] ;

(v14-3) [v:ghost,position=270:10mm from v13-3,color=Gray] ;
(v14-4) [v:ghost,position=0:15mm from v14-3,color=Gray] ;
(s14-3) [v:ghost,position=270:5mm from v14-3,color=Gray,opacity=0.35] ;
(s14-4) [v:ghost,position=180:6.5mm from v14-4,color=Gray,opacity=0.35] ;
(v14-5) [v:main,position=0:11mm from v14-4,color=Gray] ;
(w14-5) [v:main,position=0:5mm from v14-5,color=Gray,fill=white] ;
(v14-6) [v:main,position=0:13mm from v14-5,color=Gray] ;
(w14-6) [v:main,position=0:5mm from v14-6,color=Gray,fill=white] ;
(v14-7) [v:main,position=0:13mm from v14-6,color=Gray] ;
(w14-7) [v:main,position=0:5mm from v14-7,color=Gray,fill=white] ;
(v14-8) [v:main,position=0:13mm from v14-7,color=Gray] ;
(w14-8) [v:main,position=0:5mm from v14-8,color=Gray,fill=white] ;
(v14-9) [v:main,position=0:13mm from v14-8,color=Gray] ;
(w14-9) [v:main,position=0:5mm from v14-9,color=Gray,fill=white] ;
(v14-10) [v:main,position=0:13mm from v14-9,color=Gray] ;
(w14-10) [v:main,position=0:5mm from v14-10,color=Gray,fill=white] ;
(v14-11) [v:main,position=0:13mm from v14-10,color=Gray] ;
(w14-11) [v:main,position=0:5mm from v14-11,color=Gray,fill=white] ;
(v14-12) [v:main,position=0:13mm from v14-11,color=Gray] ;
(w14-12) [v:main,position=0:5mm from v14-12,color=Gray,fill=white] ;
(v14-13) [v:main,position=0:13mm from v14-12,color=Gray] ;
(w14-13) [v:main,position=0:5mm from v14-13,color=Gray,fill=white] ;
(v14-14) [v:main,position=0:13mm from v14-13,color=Gray] ;
(w14-14) [v:main,position=0:5mm from v14-14,color=Gray,fill=white] ;

(centre) [fill=BritishRacingGreen,minimum width=12.5mm,minimum height=5mm,opacity=0.3,position=270:5mm from v88] ;

[e:coloredborder,color=Gray] (s34) to (t34);
[e:coloredborder,color=Gray] (v35) to (w35);
[e:coloredborder,color=Gray] (v36) to (w36);
[e:coloredborder,color=Gray] (v37) to (w37);
[e:coloredborder,color=Gray] (v38) to (w38);
[e:coloredborder,color=Gray] (v39) to (w39);
[e:coloredborder,color=Gray] (v310) to (w310);
[e:coloredborder,color=Gray] (v311) to (w311);
[e:coloredborder,color=Gray] (v312) to (w312);
[e:coloredborder,color=Gray] (v313) to (w313);
[e:coloredborder,color=Gray] (v314) to (w314);

[e:colored,line width=1.7pt,color=BostonUniversityRed] (s34) to (t34);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v35) to (w35);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v36) to (w36);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v37) to (w37);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v38) to (w38);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v39) to (w39);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v310) to (w310);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v311) to (w311);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v312) to (w312);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v313) to (w313);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v314) to (w314);

[e:coloredborder,color=Gray] (s53) to (t53);
[e:coloredborder,color=Gray] (s54) to (t54);
[e:coloredborder] (v55) to (w55);
[e:coloredborder] (v56) to (w57);
[e:coloredborder] (v57) to (w58);
[e:coloredborder] (v58) to (w59);
[e:coloredborder] (v59) to (w510);
[e:coloredborder] (v510) to (w511);
[e:coloredborder] (v511) to (w512);
[e:coloredborder] (v512) to (w513);
[e:coloredborder,color=Gray] (v513) to (w514);
[e:coloredborder,color=Gray] (v514) to (w515);

[e:colored,line width=1.7pt,color=BostonUniversityRed] (s53) to (t53);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (s54) to (t54);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v55) to (w55);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v56) to (w57);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v57) to (w58);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v58) to (w59);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v59) to (w510);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v510) to (w511);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v511) to (w512);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v512) to (w513);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v513) to (w514);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v514) to (w515);

[e:coloredborder,color=Gray] (s73) to (t73);
[e:coloredborder,color=Gray] (s74) to (t74);
[e:coloredborder] (v75) to (w75);
[e:coloredborder] (v76) to (w77);
[e:coloredborder] (v77) to (w78);
[e:coloredborder] (v78) to (w79);
[e:coloredborder] (v79) to (w710);
[e:coloredborder] (v710) to (w711);
[e:coloredborder] (v711) to (w712);
[e:coloredborder] (v712) to (w713);
[e:coloredborder,color=Gray] (v713) to (w714);
[e:coloredborder,color=Gray] (v714) to (w715);

[e:colored,line width=1.7pt,color=BostonUniversityRed] (s73) to (t73);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (s74) to (t74);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v75) to (w75);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v76) to (w77);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v77) to (w78);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v78) to (w79);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v79) to (w710);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v710) to (w711);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v711) to (w712);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v712) to (w713);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v713) to (w714);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v714) to (w715);

[e:coloredborder,color=Gray] (s93) to (t93);
[e:coloredborder,color=Gray] (s94) to (t94);
[e:coloredborder] (v95) to (w95);
[e:coloredborder] (v96) to (w97);
[e:coloredborder,color=BritishRacingGreen] (v97) to (w98);
[e:coloredborder,color=BritishRacingGreen] (v98) to (w99);
[e:coloredborder] (v99) to (w910);
[e:coloredborder] (v910) to (w911);
[e:coloredborder] (v911) to (w912);
[e:coloredborder] (v912) to (w913);
[e:coloredborder,color=Gray] (v913) to (w914);
[e:coloredborder,color=Gray] (v914) to (w915);

[e:colored,line width=1.7pt,color=BostonUniversityRed] (s93) to (t93);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (s94) to (t94);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v95) to (w95);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v96) to (w97);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v97) to (w98);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v98) to (w99);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v99) to (w910);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v910) to (w911);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v911) to (w912);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v912) to (w913);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v913) to (w914);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v914) to (w915);

[e:coloredborder,color=Gray] (s11-3) to (t11-3);
[e:coloredborder,color=Gray] (s11-4) to (t11-4);
[e:coloredborder] (v11-5) to (w11-5);
[e:coloredborder] (v11-6) to (w11-7);
[e:coloredborder] (v11-7) to (w11-8);
[e:coloredborder] (v11-8) to (w11-9);
[e:coloredborder] (v11-9) to (w11-10);
[e:coloredborder] (v11-10) to (w11-11);
[e:coloredborder] (v11-11) to (w11-12);
[e:coloredborder] (v11-12) to (w11-13);
[e:coloredborder,color=Gray] (v11-13) to (w11-14);
[e:coloredborder,color=Gray] (v11-14) to (w11-15);

[e:colored,line width=1.7pt,color=BostonUniversityRed] (s11-3) to (t11-3);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (s11-4) to (t11-4);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v11-5) to (w11-5);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v11-6) to (w11-7);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v11-7) to (w11-8);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v11-8) to (w11-9);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v11-9) to (w11-10);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v11-10) to (w11-11);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v11-11) to (w11-12);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v11-12) to (w11-13);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v11-13) to (w11-14);
[e:colored,line width=1.7pt,color=BostonUniversityRed] (v11-14) to (w11-15);

[e:coloredborder,color=Gray] (s13-3) to (t13-3);