The polyomino graphs whose resonance graphs have a 1-degree vertex

Abstract

A polyomino graph is a 2-connected plane graph such that each interior face is a square of side length one. The resonance graph of a polyomino graph P is a graph in which the vertices are the 1-factors of P and two vertices are adjacent provided their corresponding two 1-factors differ in exactly one unite square. In this paper, we determine the polyomino graphs whose resonance graph has a 1-degree vertex by constructing a 4-coordinate system. As consequences we determine the polyomino graphs with a forcing edge and an anti-forcing edge respectively.