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

Let G = (V(G), E(G)) be a graph. For x E 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] n 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 yiD(G). In this paper, we study the identifying code of the direct product Kr, x G, where K is a complete graph. We first give a lower bound of ylD(K, x G) when G is a regular graph. Then we compute yID(K, x G) for G being a cycle and a complete bipartite graph, respectively. (C) 2018 Elsevier B.V. All rights reserved.