Extremal anti-forcing numbers of perfect matchings of graphs

TL;DR

This paper studies the anti-forcing numbers of perfect matchings in graphs, characterizes extremal cases, and provides efficient algorithms for specific graph classes, advancing understanding of perfect matching uniqueness.

Contribution

It characterizes graphs with minimal and maximal anti-forcing numbers, and determines the anti-forcing spectrum of even polygonal chains efficiently.

Findings

Minimum anti-forcing number is one for certain plane bipartite graphs.

Maximum anti-forcing number is bounded by the graph's cyclomatic number.

Anti-forcing spectrum of even polygonal chains can be computed in linear time.

Abstract

The anti-forcing number of a perfect matching $M$ of a graph $G$ is the minimal number of edges not in $M$ whose removal to make $M$ as a unique perfect matching of the resulting graph. The set of anti-forcing numbers of all perfect matchings of $G$ is the anti-forcing spectrum of $G$. In this paper, we characterize the plane elementary bipartite graph whose minimum anti-forcing number is one. We show that the maximum anti-forcing number of a graph is at most its cyclomatic number. In particular, we characterize the graphs with the maximum anti-forcing number achieving the upper bound, such extremal graphs are a class of plane bipartite graphs. Finally, we determine the anti-forcing spectrum of an even polygonal chain in linear time.