Identifying codes of corona product graphs

TL;DR

This paper investigates the properties of identifying codes in corona product graphs, providing conditions for their existence and formulas to compute their minimum size based on component graphs.

Contribution

It establishes necessary and sufficient conditions for the identifiability of corona product graphs and expresses the minimum identifying code size in terms of component graph parameters.

Findings

Provides a characterization for identifiable corona product graphs.

Derives a formula for the minimum identifying code size in corona products.

Calculates identifying code sizes for specific classes of graphs.

Abstract

For a vertex $x$ of a graph $G$, let $N_G[x]$ be the set of $x$ with all of its neighbors in $G$. A set $C$ of vertices is an {\em identifying code} of $G$ if the sets $N_G[x]\cap C$ are nonempty and distinct for all vertices $x$. If $G$ admits an identifying code, we say that $G$ is identifiable and denote by $\gamma^{ID}(G)$ the minimum cardinality of an identifying code of $G$. In this paper, we study the identifying code of the corona product $H\odot G$ of graphs $H$ and $G$. We first give a necessary and sufficient condition for the identifiable corona product $H\odot G$, and then express $\gamma^{ID}(H\odot G)$ in terms of $\gamma^{ID}(G)$ and the (total) domination number of $H$. Finally, we compute $\gamma^{ID}(H\odot G)$ for some special graphs $G$.