The distinguishing number and the distinguishing index of co-normal product of two graphs

TL;DR

This paper investigates the distinguishing number and index of the co-normal product of graphs, proving that certain powers of connected graphs without false twin or dominating vertices have these parameters equal to two.

Contribution

It establishes that for any connected graph without false twin or dominating vertices, all higher co-normal powers have a distinguishing number and index of two, extending understanding of graph symmetries.

Findings

For all k ≥ 3, the k-th co-normal power of such graphs has D(G)=2 and D'(G)=2.

The paper characterizes the automorphism groups of co-normal products.

It provides conditions under which the distinguishing parameters are minimized.

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 co-normal product $G\star H$ of two graphs $G$ and $H$ is the graph with vertex set $V (G)\times V (H)$ and edge set $\{\{(x_1, x_2), (y_1, y_2)\} | x_1y_1 \in E(G) ~{\rm or}~x_2y_2 \in E(H)\}$. In this paper we study the distinguishing number and the distinguishing index of the co-normal product of two graphs. We prove that for every $k \geq 3$, the $k$-th co-normal power of a connected graph $G$ with no false twin vertex and no dominating vertex, has the distinguishing number and the distinguishing index equal two.