# Distinguishing number and distinguishing index of strong product of two graphs

## TL;DR

This paper investigates the distinguishing number and index of the strong product of two graphs, providing new bounds and properties, especially for the strong powers of connected S-thin graphs, with a focus on automorphism-breaking labelings.

## Contribution

It establishes that for any connected S-thin graph, all its strong powers have a distinguishing index of 2, advancing understanding of symmetry breaking in graph products.

## Key findings

- Strong powers of connected S-thin graphs have distinguishing index 2.
- Provides bounds for the distinguishing number of strong products.
- Analyzes automorphism-preserving labelings in graph products.

## 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 strong product $G\boxtimes H$ of two graphs $G$ and $H$ is the graph with vertex set $V (G)\times V (H)$ and edge set $\{\{(x_1, x_2), (y_1, y_2)\} | x_iy_i \in E(G_i) ~{\rm or}~ x_i = y_i ~{\rm for~ each}~ 1 \leq i \leq 2.\}$. In this paper we study the distinguishing number and the distinguishing index of strong product of two graphs. We prove that for every $k \geq 2$, the $k$-th strong power of a connected $S$-thin graph $G$ has distinguishing index equal 2.