Identifying Codes and Domination in the Product of Graphs

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. The minimum cardinality of an identifying code in a graph GG is denoted \gid(G). We consider identifying codes of the direct product Kn×KmK_n \times K_m. In particular, we answer a question of Klav\v{z}ar and show the exact value of \gid(K_n \times K_m). It was recently shown by Gravier, Moncel and Semri that for the Cartesian product of two same-sized cliques \gid(K_n \Box K_n) = \lfloor{\frac{3n}{2}\rfloor}. Letting mn2m \ge n \ge 2 be any integers, we show that \IDCODE(K_n \cartprod K_m) = \max\{2m-n, m + \lfloor n/2 \rfloor\}. Furthermore, we improve upon the bounds for \IDCODE(G \cartprod K_m) and explore the specific case when GG is the Cartesian product of multiple cliques. Given two disjoint copies of a graph GG, denoted G1G^1 and G2G^2, and a permutation π\pi of V(G)V(G), the permutation graph πG\pi G is constructed by joining uV(G1)u \in V(G^1) to π(u)V(G2)\pi(u) \in V(G^2) for all uV(G1)u \in V(G^1). The graph GG is said to be a universal fixer if the domination number of πG\pi G is equal to the domination number of GG for all π\pi of V(G)V(G). In 1999 it was conjectured that the only universal fixers are the edgeless graphs. We prove the conjecture

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