Complete forcing numbers of multiple hexagonal chains

Abstract

Let G be a graph with a perfect matching. The complete forcing number of G is the minimum cardinality of an edge subset S of G such that for each perfect matching M of G, the intersection of S and M forms a forcing set of M. Chan et al. showed that the complete forcing number of a catacondensed hexagonal system is equal to the sum of the number of hexagons and its Clar number. Closed formulas for the complete forcing numbers of certain peri-condensed hexagonal systems, such as parallelograms, have also been established. In this paper, we consider the complete forcing number of a multiple hexagonal chain (MHC), as we identified a mistake in a published result. We establish an upper bound on the complete forcing number of an MHC and a lower bound on that of a normal hexagonal system via face coloring. Using these bounds, we obtain explicit expressions for the complete forcing numbers of MHCs with 3m (m is a positive integer) columns of hexagons, zigzag MHCs and chevrons.