Identifying codes of Cartesian product of two cliques

TL;DR

This paper generalizes the known results on the minimum size of identifying codes in Cartesian products of two equal-sized cliques to the case where the cliques may differ in size, expanding the understanding of identifying codes in these graph structures.

Contribution

It extends the previous work by Gravier et al. to include Cartesian products of any two nontrivial cliques, providing a broader characterization of identifying codes.

Findings

Generalized the minimum cardinality of identifying codes for Cartesian products of two nontrivial cliques.

Provided explicit formulas or bounds for different clique size combinations.

Enhanced the theoretical understanding of identifying codes in complex graph products.

Abstract

An identifying code in a graph $G$ is a dominating set $C$ such that the closed neighborhood of each vertex in $G$ has a distinct intersection with $C$. In 2008, Gravier et al. determined the minimum cardinality of an identifying code of the Cartesian product of two cliques with the same size. In this note, we generalize this result to the Cartesian product of any two nontrivial cliques.