Identifying Codes and Domination in the Product of Graphs

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(G). We consider identifying codes of the direct product Kn × Km. In particular, we answer a question of Klavžar and show the exact value of γ(Kn×Km). It was recently shown by Gravier, Moncel and Semri that for the Cartesian product of two same-sized cliques γ(Kn Kn) = b 3n 2 c. Letting m ≥ n ≥ 2 be any integers, we show that γ(Kn Km) = max{2m − n,m + bn/2c}. Furthermore, we improve upon the bounds for γ(G Km) and explore the specific case when G is the Cartesian product of multiple cliques. Given two disjoint copies of a graph G, denoted G and G, and a permutation π of V (G), the permutation graph πG is constructed by joining u ∈ V (G) to π(u) ∈ V (G) for all u ∈ V (G). The graph G is said to be a universal fixer if the domination number of πG is equal to the domination number of G for all π of V (G). In 1999 it was conjectured that the only universal fixers are the edgeless graphs. We prove the conjecture.