Complete forcing numbers of catacondensed hexagonal systems

Let G be a graph with 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. A global forcing set of $$G$$G, introduced by Vukičević et al., is a subset of $$E(G)$$E(G) on which there are distinct restrictions of any two different perfect matchings of $$G$$G. Combining the above “forcing” and “global” ideas, we introduce and define a complete forcing set of G as a subset of $$E(G)$$E(G) on which the restriction of any perfect matching $$M$$M of $$G$$G is a forcing set of $$M$$M. The minimum cardinality of complete forcing sets is the complete forcing number of $$G$$G. First we establish some initial results about these two novel concepts, including a criterion for a complete forcing set, and comparisons between the complete forcing number and global forcing number. Then we give an explicit formula for the complete forcing number of a hexagonal chain. Finally a recurrence relation for the complete forcing number of a catacondensed hexagonal system is derived.