Reconfiguration graphs of zero forcing sets

TL;DR

This paper introduces the concept of zero forcing graphs, explores their properties for different base graphs, and analyzes the computational complexity of constructing these graphs.

Contribution

It defines zero forcing graphs, characterizes their structure for specific graphs, and analyzes the complexity of computing them.

Findings

Zero forcing graph of a forest is connected.

Many zero forcing graphs are disconnected.

Computing the zero forcing graph takes exponential time in the worst case.

Abstract

This paper begins the study of reconfiguration of zero forcing sets, and more specifically, the zero forcing graph. Given a base graph $G$, its zero forcing graph, $\mathscr{Z}(G)$, is the graph whose vertices are the minimum zero forcing sets of $G$ with an edge between vertices $B$ and $B'$ of $\mathscr{Z}(G)$ if and only if $B$ can be obtained from $B'$ by changing a single vertex of $G$. It is shown that the zero forcing graph of a forest is connected, but that many zero forcing graphs are disconnected. We characterize the base graphs whose zero forcing graphs are either a path or the complete graph, and show that the star cannot be a zero forcing graph. We show that computing $\mathscr{Z}(G)$ takes $2^{\Theta(n)}$ operations in the worst case for a graph $G$ of order $n$.