Reversing monoid actions and domination in graphs

TL;DR

This paper introduces a novel approach using reversing monoid actions on graph dynamical systems to identify special dominating and nonblocking vertex sets, providing new insights into graph domination properties.

Contribution

It presents a new method of reversing graph dynamical system actions to detect and analyze dominating and nonblocking sets, linking these concepts to free monoid actions.

Findings

Reversing the action of graph dynamical systems reveals special dominating sets.

Multiple system actions define free monoid actions related to dominating sets.

The approach offers a new perspective on graph domination via algebraic dynamical systems.

Abstract

Given a graph $G=(V,E)$, a set of vertices $D\subseteq V $ is called a dominating set if every vertex in $V\backslash D$ is adjacent to a vertex in $D$, and a subset $B\subseteq V $ is called a nonblocking set if $V-B$ is a dominating set. In this paper, we introduce a graph dynamical systems detecting vertex sets that are simultaneously dominating and nonblocking sets via reversing the action of the system. Moreover, by using actions of multiple such graph dynamical systems we define actions of free monoid on two letters for which elements in the reverse action corresponds to more special dominating sets.