Identifying codes of the direct product of two cliques

TL;DR

This paper determines the minimum size of identifying codes in the direct product of two cliques, extending previous work on Cartesian products and answering an open question in graph theory.

Contribution

It provides a complete characterization of the minimum identifying code size for the direct product of any two cliques, a problem previously unresolved.

Findings

Minimum identifying code size for the direct product of two cliques is established.

The result generalizes known cases for Cartesian products.

The paper answers an open question posed by Klavzar.

Abstract

An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. It was recently shown by Gravier, Moncel and Semri that the minimum cardinality of an identifying code for the Cartesian product of two cliques of the same order n is the floor of 3n/2. We consider identifying codes of the direct product of two cliques. In particular, we answer a question of Klavzar and determine the minimum cardinality of an identifying code for the direct product of any two cliques.