Elementary components of essentially disconnected polyomino graphs

Abstract

An essentially disconnected polyomino graph is defined as a polyomino graph with some perfect matchings and forbidden edges. In this paper, we prove that the subgraph, obtained by deleting all the forbidden edges, is disconnected and has at least two elementary components, which generalizes the results for essentially disconnected benzenoid systems by Gutman et al. Furthermore, we show that if an essentially disconnected polyomino graph has an unit square as one of its elementary components, then it has at least three elementary components.