Distinguishing number and distinguishing index of lexicographic product of two graphs

TL;DR

This paper investigates the distinguishing number and index of lexicographic graph products, establishing bounds and properties, including how powers of such graphs can be distinguished with minimal labels.

Contribution

It provides sharp bounds for the distinguishing number and index of lexicographic graph products and explores their behavior under graph powers, including minimal label distinctions.

Findings

Bounds for D(G[H]) and D'(G[H]) are established.

For connected graphs with certain automorphism conditions, D(G^k) is bounded between D(G) and D(G)+k-1.

All lexicographic powers G^k can be distinguished with at most two labels.

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. The lexicographic product of two graphs $G$ and $H$, $G[H]$ can be obtained from $G$ by substituting a copy $H_u$ of $H$ for every vertex $u$ of $G$ and then joining all vertices of $H_u$ with all vertices of $H_v$ if $uv\in E(G)$. In this paper we obtain some sharp bounds for the distinguishing number and the distinguishing index of lexicographic product of two graphs. As consequences, we prove that if $G$ is a connected graph with a special condition on automorphism group of $G[G]$ and $D(G)> 1$, then for every natural $k$, $D(G)\leq D(G^k)\leq D(G)+k-1$, where $G^k=G[G[...]]$. Also we prove that all lexicographic powers of $G$, $G^k$ ($k\geq 2$) can be distinguished by at most two edge labels.