Complete Forcing Numbers of Random Multiple Hexagonal Chains

Abstract

Let G be a simple connected graph with vertex set V(G) and edge set E(G) that admits a perfect matching M. A forcing set of M is a subset of M contained in no other perfect matchings of G. The minimum cardinality of forcing sets is the forcing number of M. A complete forcing set of G, recently introduced by Xu et al. [Complete forcing numbers of catacondensed hexagonal systems, J. Combin. Optim. 29(4) (2015) 803-814], is a subset S of E(G) on which the restriction of any perfect matching M of G is a forcing set of M. A complete forcing set of the smallest cardinality is called a minimum complete forcing set, and its cardinality is the complete forcing number of G, denoted by cf(G). In this paper, we present the complete forcing sets and complete forcing number of random multiple hexagonal chains.