Complete Forcing Numbers of Random Multiple Hexagonal Chains

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