Complete forcing numbers of hexagonal systems

Abstract

As a strengthening of the concept of global forcing number of a graph G, the complete forcing number of G is the cardinality of a minimum edge subset of G to which the restriction of every perfect matching M is a forcing set of M. Xu et al. (J Comb Opt 29: 803–814, 2015) revealed that a complete forcing set of G also antifores each perfect matching, and obtained that for a catacondensed hexagonal system, the complete forcing number is equal to the Clar number plus the number of hexagons (Chan et al. MATCH Commun Math Comput Chem 74: 201–216, 2015). In this paper, we consider general hexagonal systems H, and present sharp upper bound on the complete forcing number of H in terms of elementary edge-cut cover and lower bound via graph decomposition as well. Through such approaches, we obtain some closed formulas for the complete forcing numbers of some types of hexagonal systems including parallelogram, regular hexagon- and rectangle-shaped hexagonal systems.