Some tight bounds on the minimum and maximum forcing numbers of graphs

Let G be a simple graph with 2n vertices and a perfect matching. We denote by f (G) and F (G) the minimum and maximum forcing numbers of G, respectively. Hetyei obtained that the number of edges of graphs G with a unique perfect matching is at most n2. Since a graph G has a unique perfect matching if and only if f (G) = 0, along this line, we generalize easily the classical result to all graphs G with f (G) = k for 0 <= k <= n- 1, and get a non-trivial lower bound of f (G) in terms of the order and size. For bipartite graphs, we gain the corresponding stronger results. Such lower bounds enable one to obtain the minimum forcing number of some dense graphs. Further, we obtain a new upper bound of F (G). For bipartite graphs G, Che and Chen (2013) obtained that f (G) = n - 1 if and only if G is complete bipartite graph Kn,n. We completely characterize all bipartite graphs G with f (G) = n - 2. (c) 2022 Elsevier B.V. All rights reserved.