Continuous forcing spectrum of regular hexagonal polyhexes

Abstract

For any perfect matching M of a graph G, the forcing number (resp. anti-forcing number) of M is the smallest cardinality of an edge subset S⊆M (resp. S⊆E(G)∖M) such that the graph G−V(S) (resp. G−S) has a unique perfect matching. The forcing spectrum of G is the set of forcing numbers of all perfect matchings of G. Afshani et al. [2] proved that any finite set of positive integers can be the forcing spectrum of a planar bipartite graph. A polyhex is a 2-connected plane bipartite graph whose interior faces are regular hexagons. In this paper, we give the minimum forcing number and anti-forcing number of a polyhex with a 3-divisible perfect matching. Further, we prove that the forcing spectrum of any regular hexagonal polyhex is continuous, i.e. an integer interval, by applying some changes on prolate triangle polyhexes to obtain a sequence of perfect matchings of the graph.