A polyomino graph P is a connected finite subgraph of the infinite plane grid such that each finite face is surrounded by a regular square of side length one and each edge belongs to at least one square. A dimer covering of P corresponds to a perfect matching. Different dimer coverings can interact via an alternating cycle (or square) with respect to them. A set of disjoint squares of P is a resonant set if P has a perfect matching M so that each one of those squares is M-alternating. In this paper, we show that if K is a maximum resonant set of P, then P − K has a unique perfect matching. We further prove that the maximum forcing number of a polyomino graph is equal to the cardinality of a maximum resonant set. This confirms a conjecture of Xu et al. [26]. We also show that if K is a maximal alternating set of P, then P − K has a unique perfect matching.
@article{bwmeta1.element.doi-10_7151_dmgt_1857,
author = {Heping Zhang and Xiangqian Zhou},
title = {A Maximum Resonant Set of Polyomino Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {36},
year = {2016},
pages = {323-337},
zbl = {1334.05122},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1857}
}
Heping Zhang; Xiangqian Zhou. A Maximum Resonant Set of Polyomino Graphs. Discussiones Mathematicae Graph Theory, Tome 36 (2016) pp. 323-337. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1857/