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

The maximum forcing number of polyomino