Relations between global forcing number and maximum anti-forcing number of a graph

Abstract

The global forcing number of a graph G is the minimal cardinality of an edge subset discriminating all perfect matchings of G, denoted by gf(G). For a perfect matching M of G, the minimal cardinality of an edge subset S⊆E(G)∖M such that G−S has a unique perfect matching is called the anti-forcing number of M. The maximum anti-forcing number of G among all perfect matchings is denoted by Af(G). It is known that the maximum anti-forcing number of a hexagonal system equals the famous Fries number.For a bipartite graph G, we show that gf(G)≥Af(G). Next we extend such result to Birkhoff–von Neumann graphs, whose perfect matching polytopes are characterized solely by nonnegativity and degree constraints, by revealing an odd dumbbell of non-bipartite graphs with a unique perfect matching and minimum degree at least two. Finally, we obtain tight upper and lower bounds on gf(G)−Af(G). For a connected bipartite graph G with 2n vertices, 0≤gf(G)−Af(G)≤12(n−1)(n−2). For non-bipartite case, we have −Occ(G)≤gf(G)−Af(G)≤(n−1)(n−2) by introducing a new nonnegative parameter Occ(G).