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

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 subset of 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) <= 1/2 (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). (c) 2022 Elsevier B.V. All rights reserved.