Maximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of Graphs

TL;DR

This paper characterizes graphs with maximum forcing number of perfect matchings as complete multipartite graphs or certain modifications of complete bipartite graphs, and analyzes the spectrum of forcing numbers.

Contribution

It completely solves the open problem of characterizing graphs with maximum forcing number, extending previous bipartite results to broader classes of graphs.

Findings

Graphs with maximum forcing number are complete multipartite or derived from complete bipartite graphs.

For graphs with maximum forcing number, the forcing spectrum forms an integer interval.

Minimum forcing numbers range from half of n to n-1, forming an integer interval.

Abstract

Let $G$ be a simple graph with $2n$ vertices and a perfect matching. The forcing number $f(G,M)$ of a perfect matching $M$ of $G$ is the smallest cardinality of a subset of $M$ that is contained in no other perfect matching of $G$. Among all perfect matchings $M$ of $G$, the minimum and maximum values of $f(G,M)$ are called the minimum and maximum forcing numbers of $G$, denoted by $f(G)$ and $F(G)$, respectively. Then $f(G)\leq F(G)\leq n-1$. Che and Chen (2011) proposed an open problem: how to characterize the graphs $G$ with $f(G)=n-1$. Later they showed that for a bipartite graph $G$, $f(G)=n-1$ if and only if $G$ is a complete bipartite graph $K_{n,n}$. In this paper, we completely solve the problem of Che and Chen, and show that $f(G)=n-1$ if and only if $G$ is a complete multipartite graph or a graph obtained from complete bipartite graph $K_{n,n}$ by adding arbitrary edges in the same partite set. For all graphs $G$ with $F(G)=n-1$, we prove that the forcing spectrum of each such graph $G$ forms an integer interval by matching 2-switches and the minimum forcing numbers of all such graphs $G$ form an integer interval from $\lfloor\frac{n}{2}\rfloor$ to $n-1$.