Identifying codes of the direct product of two cliques

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. The minimum cardinality of an identifying code in a graph G is denoted γ ID ( G ) . It was recently shown by Gravier, Moncel and Semri that γ ID ( K n □ K n ) = ⌊ 3 n 2 ⌋ . Letting n , m ≥ 2 be any integers, we consider identifying codes of the direct product K n × K m . In particular, we answer a question of Klavžar and show the exact value of γ ID ( K n × K m ) .