called vertical edges) and $k$ horizontal edges from $i_{2u}$ to $i_{2u}+s$, $k$ horizontal edges from $i_{2u-1}+s$ to $i_{2u-1}$ (with the convention that $i_{2k+1}=i_1$), where all the $u$’s are in the region: $1\leq u\leq k$. An example of a $Q$ graph with $k=3$ is shown in Figure \[sixq\].

![An example of a $Q$ graph with $k=3$.[]{data-label="sixq"}](sixq.eps){width="13cm"}

The subgraph of all I-vertices is called the **roof** of $Q$ and is denoted by $H(Q)$ (see also Figure \[sixq\] for illustration of $H(Q)$).

The M-**minor or pillar** of $Q$ is defined as the **minor** of $Q$ by contracting all horizontal edges, which means that all horizontal edges are removed from Q and all I-vertices connected through horizontal edges are glued together. We denote the M-**minor or pillar** of $Q$ by $M(Q)$ (see Figure \[sixm\]).

![The M-minor or pillar $M(Q)$ that corresponds to the graph $Q$ in Figure \[sixq\].[]{data-label="sixm"}](sixm.eps){width="11cm"}

For a given $M(Q)$, glue all coincident vertical edges; namely, we regard all vertical edges with a common I-vertex and J-vertex as one edge. Then we get an undirectional connected graph. We call the resulting graph the **base** of the graph Q, and denote it by B(Q) (see Figure \[sixb\]).

![The base $B(Q)$ that corresponds to 