[<span style="font-variant:small-caps;">B$_1$-VPG</span> ]{}or [<span style="font-variant:small-caps;">B$_1$-EPG</span> ]{}graph as input (in addition to $G$). We sometimes violate the wording and say *path(s) in $G$* to actually refer to the vertices in $G$ corresponding to the paths in [$\mathcal{P}_{{\texttt{vtx}}}$ ]{}or [$\mathcal{P}_{{\texttt{edg}}}$ ]{}. We denote the $x$- and $y$-coordinates of a point $p$ in the plane by $x(p)$ and $y(p)$, respectively.

Let $G=(V, E)$ be a graph with its representations $\langle$[$\mathcal{P}_{{\texttt{vtx}}}$ ]{}$,\mathcal{G}\rangle$ and $\langle$[$\mathcal{P}_{{\texttt{edg}}}$ ]{}$,\mathcal{G}\rangle$, and let $P\in\{$[$\mathcal{P}_{{\texttt{vtx}}}$ ]{}$\cup$[$\mathcal{P}_{{\texttt{edg}}}$ ]{}$\}$ be an arbitrary path. Since $P$ has at most one bend, it is in one of the types {[$\ulcorner$]{}, [$\llcorner$]{}, [$\urcorner$]{}, [$\lrcorner$]{}, $|$, $-$}. We call a path $P$ of type $\texttt{x}\in${[$\ulcorner$]{}, [$\llcorner$]{}, [$\urcorner$]{}, [$\lrcorner$]{}}, a *$\texttt{x}$-type path*; we complete the definition by referring to no-bend paths as [$\llcorner$]{}-type paths.

We denote the horizontal and vertical segments of $P$ by ${\texttt{hPart}}(P)$ and ${\texttt{vPart}}(P)$, respectively. We call the common endpoint of ${\texttt{hPart}}(P)$ and ${\texttt{vPart}}(P)$ the *corner* of $P$ and denote it by ${\texttt{corner}}(P)$. Moreover, let ${\texttt{hTip}}(P)$ (resp., ${\texttt{vTip}}(P)$) denote the endpoint of ${\texttt{hPart}}(P)$ (resp., ${\texttt{vPart}}(P)$) that is not shared with ${\texttt{vPart}}(P)$ (resp., not shared with ${\texttt{hPart}}(P)$). Let $N[P]$ denote the set of paths adjacent to $P$; we assume that $P\in N[P]$. Moreover, for a set 