The maximum forcing number of polyomino

The forcing number of a perfect matching $M$ of a graph $G$ is the cardinality of the smallest subset of $M$ that is contained in no other perfect matchings of $G$. For a planar embedding of a 2-connected bipartite planar graph $G$ which has a perfect matching, the concept of Clar number of hexagonal system had been extended by Abeledo and Atkinson as follows: a spanning subgraph $C$ of is called a Clar cover of $G$ if each of its components is either an even face or an edge, the maximum number of even faces in Clar covers of $G$ is called Clar number of $G$, and the Clar cover with the maximum number of even faces is called the maximum Clar cover. It was proved that if $G$ is a hexagonal system with a perfect matching $M$ and $K'$ is a set of hexagons in a maximum Clar cover of $G$, then $G-K'$ has a unique 1-factor. Using this result, Xu {\it et. at.} proved that the maximum forcing number of the elementary hexagonal system are equal to their Clar numbers, and then the maximum forcing number of the elementary hexagonal system can be computed in polynomial time. In this paper, we show that an elementary polyomino has a unique perfect matching when removing the set of tetragons from its maximum Clar cover. Thus the maximum forcing number of elementary polyomino equals to its Clar number and can be computed in polynomial time. Also, we have extended our result to the non-elementary polyomino and hexagonal system

Tight upper bound on the maximum anti-forcing numbers of graphs

Let $G$ be a simple graph with a perfect matching. Deng and Zhang showed that the maximum anti-forcing number of $G$ is no more than the cyclomatic number. In this paper, we get a novel upper bound on the maximum anti-forcing number of $G$ and investigate the extremal graphs. If $G$ has a perfect matching $M$ whose anti-forcing number attains this upper bound, then we say $G$ is an extremal graph and $M$ is a nice perfect matching. We obtain an equivalent condition for the nice perfect matchings of $G$ and establish a one-to-one correspondence between the nice perfect matchings and the edge-involutions of $G$, which are the automorphisms $\alpha$ of order two such that $v$ and $\alpha(v)$ are adjacent for every vertex $v$. We demonstrate that all extremal graphs can be constructed from $K_2$ by implementing two expansion operations, and $G$ is extremal if and only if one factor in a Cartesian decomposition of $G$ is extremal. As examples, we have that all perfect matchings of the complete graph $K_{2n}$ and the complete bipartite graph $K_{n, n}$ are nice. Also we show that the hypercube $Q_n$, the folded hypercube $FQ_n$ ($n\geq4$) and the enhanced hypercube $Q_{n, k}$ ($0\leq k\leq n-4$) have exactly $n$, $n+1$ and $n+1$ nice perfect matchings respectively.Comment: 15 pages, 7 figure