The distinguishing number (index) and the domination number of a graph

TL;DR

This paper explores bounds on the distinguishing number and index of a graph using its domination number, providing insights into graph symmetry and domination properties.

Contribution

It introduces new upper bounds for the distinguishing number and index based on the graph's domination number.

Findings

Derived upper bounds for D(G) and D'(G) using domination number γ(G)

Established relationships between symmetry-breaking parameters and domination properties

Enhanced understanding of graph automorphisms in relation to domination sets

Abstract

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. A set $S$ of vertices in $G$ is a dominating set of $G$ if every vertex of $V(G)\setminus S$ is adjacent to some vertex in $S$. The minimum cardinality of a dominating set of $G$ is the domination number of $G$ and denoted by $\gamma (G)$. In this paper, we obtain some upper bounds for the distinguishing number and the distinguishing index of a graph based on its domination number.