On forcing matching number of boron–nitrogen fullerene graphs

Abstract

Let G be a graph that admits a perfect matching M . A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G . The smallest cardinality of forcing sets of M is called the forcing number of M . Computing the minimum forcing number of perfect matchings of a graph is an NP-complete problem. In this paper, we consider boron–nitrogen (BN) fullerene graphs, cubic 3-connected plane bipartite graphs with exactly six square faces and other hexagonal faces. We obtain the forcing spectrum of tubular BN-fullerene graphs with cyclic edge-connectivity 3. Then we show that all perfect matchings of any BN-fullerene graphs have the forcing number at least two. Furthermore, we mainly construct all seven BN-fullerene graphs with the minimum forcing number two.