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

TL;DR

This paper investigates bounds on the minimum and maximum forcing numbers of graphs with perfect matchings, providing characterizations and bounds for various classes including bipartite graphs.

Contribution

It generalizes classical results to graphs with specified forcing numbers, characterizes extremal graphs, and establishes new bounds for both minimum and maximum forcing numbers.

Findings

Characterized graphs with minimum forcing number $k$ for all $0 \\leq k \\leq n-1$.

Derived a non-trivial lower bound of $f(G)$ based on order and size.

Established a new upper bound for $F(G)$ and characterized bipartite graphs with $f(G)=n-2$.

Abstract

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 number of $G$, respectively. Hetyei obtained that the maximum number of edges of graphs $G$ with a unique perfect matching is $n^2$. We know that $G$ has a unique perfect matching if and only if $f(G)=0$. Along this line, we generalize the classical result to all graphs $G$ with $f(G)=k$ for $0\leq k\leq n-1$, and characterize corresponding extremal graphs as well. Hence we get a non-trivial lower bound of $f(G)$ in terms of the order and size. For bipartite graphs, we gain corresponding stronger results. 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 $K_{n,n}$. We completely characterize all bipartite graphs $G$ with $f(G)= n-2$.