A Maximum Resonant Set of Polyomino Graphs

TL;DR

This paper investigates the properties of maximum resonant sets in polyomino graphs, establishing their relation to perfect matchings and providing polynomial-time computation methods for the maximum forcing number.

Contribution

It proves that removing a maximum resonant set yields a graph with a unique perfect matching and shows the maximum forcing number equals the Clar number, enabling efficient computation.

Findings

Removing a maximum resonant set results in a graph with a unique perfect matching.

Maximum forcing number equals the Clar number for polyomino graphs.

Maximum forcing number can be computed in polynomial time.

Abstract

A polyomino graph $H$ 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. In this paper, we show that if $K$ is a maximum resonant set of $H$, then $H-K$ has a unique perfect matching. We further prove that the maximum forcing number of a polyomino graph is equal to its Clar number. Based on this result, we have that the maximum forcing number of a polyomino graph can be computed in polynomial time. We also show that if $K$ is a maximal alternating set of $H$, then $H-K$ has a unique perfect matching.