Distinguishing threshold for some graph operations

TL;DR

This paper investigates the distinguishing number and threshold for various graph operations, providing formulas that relate these parameters to the number of distinguishing colorings, enhancing understanding of graph symmetries.

Contribution

It derives formulas for the distinguishing number and threshold for vertex-sum, rooted product, corona, and lexicographic product graphs using the count of distinguishing colorings.

Findings

Formulas for the distinguishing number of specific graph operations.

Formulas for the distinguishing threshold of these graph operations.

Relation between the number of distinguishing colorings and graph symmetries.

Abstract

A vertex coloring of a graph $G$ is distinguishing if non-identity automorphisms do not preserve it. The distinguishing number, $D(G)$, is the minimum number of colors required for such a coloring and the distinguishing threshold, $\theta(G)$, is the minimum number of colors~$k$ such that any arbitrary $k$-coloring is distinguishing. Moreover, $\Phi_k (G)$ is the number of distinguishing coloring of $G$ using at most $k$ colors. In this paper, for some graph operations, namely, vertex-sum, rooted product, corona product and lexicographic product, we find formulae of the distinguishing number and threshold using $\Phi_k (G)$.