Some novel minimax results for perfect matchings of polyomino graphs

Let $G$ be a graph with a perfect matching $M$. The forcing number of $M$ in $G$, denoted by $f(G,M)$, is the minimal size of an edge subset of $M$ that are contained in no other perfect matchings of $G$, and the anti-forcing number of $M$ in $G$, denoted by $af(G,M)$, is the minimal size of an edge subset of $E\left(G\right)$ whose deletion results in a subgraph with a unique perfect matching $M$. For a polyomino graph $P$, Zhou and Zhang (2016) established a minimax result: For every perfect matching $M$ of $P$ with the maximum forcing number or minus one, $f(P,M)$ is equal to the maximum number of disjoint $M$-alternating squares in $P$. In this paper, we show that for every perfect matching $M$ of a polyomino graph $P$ which contains no 3 × 3 chessboard as a nice subgraph, $f(P,M)$ is equal to the maximum number of disjoint $M$-alternating squares in $P$. Further we show that for every perfect matching $M$ of $P$, $af(P,M)$ always equals the number of $M$-alternating squares of $P$ if and only if $P$ has no 1 × 3 chessboard as a nice subgraph.