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