On the spectrum of the forced matching number of graphs

TL;DR

This paper explores the range of possible forcing numbers for perfect matchings in various graphs, especially grid graphs, introduces new concepts for specific graph classes, and proves the computational complexity of determining the smallest forcing number.

Contribution

It studies the spectrum of forced matching numbers in grid graphs, extends the concept to other graph classes, and establishes NP-completeness of finding the minimum forcing number.

Findings

Spectrum of forced matching numbers for grid graphs analyzed

Forcing set concepts extended to other graph classes

Finding the smallest forcing number is NP-complete

Abstract

Let $G$ be a graph that admits a perfect matching. A {\sf forcing set} for a perfect matching $M$ of $G$ is a subset $S$ of $M$, such that $S$ is contained in no other perfect matching of $G$. This notion originally arose in chemistry in the study of molecular resonance structures. Similar concepts have been studied for block designs and graph colorings under the name {\sf defining set}, and for Latin squares under the name {\sf critical set}. Recently several papers have appeared on the study of forcing sets for other graph theoretic concepts such as dominating sets, orientations, and geodetics. Whilst there has been some study of forcing sets of matchings of hexagonal systems in the context of chemistry, only a few other classes of graphs have been considered. Here we study the spectrum of possible forced matching numbers for the grids $P_m \times P_n$, discuss the concept of a forcing set for some other specific classes of graphs, and show that the problem of finding the smallest forcing number of graphs is \NP--complete.