Continuous Forcing Spectra of Even Polygonal Chains

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 cardinality of a forcing set of M with the smallest size is called the forcing number of M, denoted by f(G, M). The forcing spectrum of G is defined as: Spec(G) = {f(G, M) divide M is a perfect matching of G}. In this paper, by applying the Z-transformation graph (resonance graph) we show that for any polyomino with perfect matchings and any even polygonal chain, their forcing spectra are integral intervals. Further we obtain some sharp bounds on maximum and minimum forcing numbers of hexagonal chains with given number of kinks. Forcing spectra of two extremal chains are determined.