The Interplay Between Domination and Separation in Graphs

TL;DR

This paper explores the relationships and complexities of various separation properties in graphs, focusing on how they relate to dominating sets and the interplay between a graph and its complement.

Contribution

It introduces and analyzes four separation properties, studies their complexity, and examines their relationships with domination-based codes and graph complementation.

Findings

Location and full-separation are equivalent in a graph and its complement.

Closed-separation in a graph corresponds to open-separation in its complement.

The complexity of finding minimum separating sets is analyzed.

Abstract

In the literature, several identification problems in graphs have been studied, of which, the most widely studied are the ones based on dominating sets as a tool of identification. Hereby, the objective is to separate any two vertices of a graph by their unique neighborhoods in a suitably chosen dominating or total-dominating set. Such a (total-)dominating set endowed with a separation property is often referred to as a code of the graph. In this paper, we study the four separation properties location, closed-separation, open-separation and full-separation. We address the complexity of finding minimum separating sets in a graph and study the interplay of these separation properties with several codes (establishing a particularly close relation between separation and codes based on domination) as well as the interplay of separation and complementation (showing that location and full-separation are the same on a graph and its complement, whereas closed-separation in a graph corresponds to open-separation in its complement).