The global forcing number of the parallelogram polyhex

A global forcing set in a simple connected graph GG with a perfect matching is any subset SS of E(G)E(G) such that the restriction of the characteristic function of perfect matchings of GG on SS is an injection. The number of edges in a global forcing set of the smallest cardinality is called the global forcing number of GG. In this paper we prove that for a parallelogram polyhex with mm rows and nn columns of hexagons (m≤nm≤n) the global forcing number equals m(n+1)/2m(n+1)/2 if mm is even, and n(m+1)/2n(m+1)/2 if mm is odd. Also, we provide an example of a minimum global forcing set.