Isomorphisms and properties of TAR reconfiguration graphs for zero forcing and other $X$-set parameters

TL;DR

This paper studies the structure and properties of reconfiguration graphs based on $X$-sets, such as zero forcing sets, revealing conditions for their isomorphisms and connectedness, and introducing the concept of $X$-irrelevant vertices.

Contribution

It characterizes when two $X$-TAR reconfiguration graphs are isomorphic and introduces $X$-irrelevant vertices to analyze these graphs, extending understanding of zero forcing and related parameters.

Findings

Isomorphism of $X$-TAR graphs corresponds to identical $X$-sets under relabeling.

Introduces $X$-irrelevant vertices to analyze graph isomorphisms.

Provides new families of graphs with improved connectedness bounds.

Abstract

An $X$-TAR (token addition/removal) reconfiguration graph has as its vertices sets that satisfy some property $X$, with an edge between two sets if one is obtained from the other by adding or removing one element. This paper considers the $X$-TAR graph for $X-$ sets of vertices of a base graph $G$ where the $X$-sets of $G$ must satisfy certain conditions. Dominating sets, power dominating sets, zero forcing sets, and positive semidefinite zero forcing sets are all examples of $X$-sets. For graphs $G$ and $G'$ with no isolated vertices, it is shown that $G$ and $G'$ have isomorphic $X$-TAR reconfiguration graphs if and only if there is a relabeling of the vertices of $G'$ such that $G$ and $G'$ have exactly the same $X$-sets. The concept of an $X$-irrelevant vertex is introduced to facilitate analysis of $X$-TAR graph isomorphisms. Furthermore, results related to the connectedness of the zero forcing TAR graph are given. We present families of graphs that exceed known lower bounds for connectedness parameters.