On the anti-forcing number of graph powers

TL;DR

This paper investigates the anti-forcing number, a measure related to perfect matchings, of the powers of graphs, providing insights into how this property evolves as graphs are iteratively connected at increasing distances.

Contribution

It introduces the study of the anti-forcing number for graph powers, extending understanding of perfect matchings in iteratively connected graphs.

Findings

Determines the anti-forcing number for specific classes of graph powers

Establishes bounds and formulas for the anti-forcing number in graph powers

Analyzes how graph powers influence the uniqueness of perfect matchings

Abstract

Let $G=(V,E)$ be a simple connected graph. A perfect matching (or Kekul\'e structure in chemical literature) of $G$ is a set of disjoint edges which covers all vertices of $G$. The anti-forcing number of $G$ is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by $af(G)$. For every $m\in\mathbb{N}$, the $m$th power of $G$, denoted by $G^m$, is a graph with the same vertex set as $G$ such that two vertices are adjacent in $G^m$ if and only if their distance is at most $m$ in $G$. In this paper, we study the anti-forcing number of the powers of some graphs.