The distinguishing number and the distinguishing index of line and graphoidal graphs

TL;DR

This paper investigates the distinguishing number and index of line graphs and graphoidal graphs derived from simple connected graphs, expanding understanding of symmetry-breaking labelings in these graph classes.

Contribution

It introduces the study of distinguishing parameters for line and graphoidal graphs, providing new insights into their automorphism-preserving labelings.

Findings

Determined bounds for the distinguishing number of line graphs.

Analyzed the distinguishing index of graphoidal graphs.

Established relationships between original graphs and their derived line and graphoidal graphs.

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 graphoidal cover of $G$ is a collection $\psi$ of (not necessarily open) paths in $G$ such that every path in $\psi$ has at least two vertices, every vertex of $G$ is an internal vertex of at most one path in $\psi$ and every edge of $G$ is in exactly one path in $\psi$. Let $\Omega(G,\psi)$ denote the intersection graph of $\psi$. A graph $H$ is called a graphoidal graph, if there exists a graph $G$ and a graphoidal cover $\psi$ of $G$ such that $H\cong \Omega (G, \psi)$. In this paper, we study the distinguishing number and the distinguishing index of the line graph and the graphoidal graph of a simple connected graph $G$.