Hongwei Wang

Let $G$ be a graph that admits a perfect matching. A forcing set for a perfect matching $M$ of $G$ is a subset $S$ of $M$, such that $S$ is not contained in any other perfect matchings of $G$. The forcing number of $M$, is defined as the number of edges in a minimum forcing set of $M$ and denoted by $f(G,M)$. The minimum forcing number of $G$, denoted by $f(G,\mathcal{M})$, is the minimum forcing number of all perfect matchings of $G$. There is some study of forcing number of hexagonal systems in the context of chemistry, but only a few other classes of graphs have been considered. Crop circles fullerenes have been discovered in the soot are presumably torusshaped \"graphitoids\". A toroidal polyhex (or toroidal graphitoid) $H(p, q, t)$ can be described by a string $(p, q, t)$ of three integers. In this work, the forcing number of a toroidal polyhex $H(p, q, t)$ is studied. We draw the conclusions:
1. If $p \leq q$, $f(G,\mathcal{M})=p;$ If $p>q,t=0,p-1,\cdots,p-q$, then $f(G,\mathcal{M})=q;$
2. When $p>q,1\leq t \leq {p-q-1}$, we present a efficient linear algorithm for computing $f(H,\mathcal{M})$, show that $f(H,\mathcal{M})$ is equal to the length of side of the biggest triangle in $H(p, q, t)$.