On the zero forcing number of the complement of graphs with forbidden subgraphs

TL;DR

This paper investigates the zero forcing number of the complement of various graph classes, extending previous results from trees to unicyclic and cactus graphs, especially those avoiding complete bipartite subgraphs.

Contribution

It provides a comprehensive analysis of the zero forcing number for complements of unicyclic and cactus graphs, broadening understanding beyond trees.

Findings

Zero forcing number for complements of unicyclic graphs characterized.

Zero forcing number for complements of cactus graphs determined.

Identified conditions under which the zero forcing number takes specific values.

Abstract

Motivated in part by an observation that the zero forcing number for the complement of a tree on $n$ vertices is either $n-3$ or $n-1$ in one exceptional case, we consider the zero forcing number for the complement of more general graphs under some conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs.