Identifying codes in the direct product of a complete graph and some special graphs

Abstract Let G = ( V ( G ) , E ( G ) ) be a graph. For x ∈ V ( G ) , let N G [ x ] be the closed neighborhood of x . A subset D of V ( G ) is called an identifying code of G if the sets N G [ x ] ∩ D are nonempty and distinct for all vertices x in G . The minimum cardinality of an identifying code in a graph G is denoted by γ I D ( G ) . In this paper, we study the identifying code of the direct product K n × G , where K n is a complete graph. We first give a lower bound of γ I D ( K n × G ) when G is a regular graph. Then we compute γ I D ( K n × G ) for G being a cycle and a complete bipartite graph, respectively.