  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.
