the graph $Q$ in Figure \[sixq\].[]{data-label="sixb"}](sixb.eps){width="11cm"}

For a vertical edge $e$ of $B(Q)$, the number of up (down) vertical edges of $Q$ coincident with $e$ is called the **up (down) multiplicity** of $e$.

The **degree** of a vertex $i_l$ is the number of edges incident to this vertex.

\[inn\] An up (down) **innovation** is an up (down) vertical edge which leads to a vertex that has not been visited before.

Two graphs are said to be **isomorphic** if one becomes the other by a suitable permutation on $(1,\cdots, T)$ and a suitable permutation on $(1,\cdots, p)$.

Define a **characteristic sequence** as $(d_1~u_1~\cdots~d_{2k}~u_{2k})$, where $\{u_1,\cdots,u_{2k}\}$ and $\{d_1,\cdots,d_{2k}\}$ are associated with the $2k$ up edges and $2k$ down edges of a pillar $M(Q)$ according to the following rule: $$\begin{aligned}
u_l=\left\{\begin{array}{cl}
1,& \text{the $l$-th up edge is an up innovation}~,\\
0,& \text{otherwise}~,
\end{array}\right.\end{aligned}$$ and $$\begin{aligned}
d_l=\left\{\begin{array}{cl}
0,& \text{the $l$-th down edge is an down innovation}~,\\
-1,& \text{otherwise}~.
\end{array}\right.\end{aligned}$$

An example of the characteristic sequence associated with the pillar in Figure \[sixm\] is given in the following Figure \[cha\] with $$\begin{aligned}
&\{u_1,u_2,u_3,u_4,u_5,u_6\}=\{\text{1,~1,~0,~0,~0,~0}\}~,\\
&\{d_1,d_2,d_3,d_4,d_5,d_6\}=\{\text{0,~0,~0,~-1,~-1,~0}\}~;\end{aligned}$$ that is, the corresponding characteristic sequence is (0 1 0 1 0 0 -1 0 -1 0 0 0). Conversely, it 