Identifying Codes in the Direct Product of a Path and a Complete Graph

Let G be a simple, undirected graph with vertex set V. For any vertex v is an element of V, the set N[v] is the vertex v and all its neighbors. A subset D subset of V (G) is a dominating set of G if for every v is an element of V (G), N[v] boolean AND D not equal null . And a subset F subset of V (G) is a separating set of G if for every distinct pair u, v is an element of V (G), N[u] boolean AND F not equal N[v] boolean AND F. An identifying code of G is a subset C subset of V (G) that is dominating as well as separating. The minimum cardinality of an identifying code in a graph G is denoted by gamma(ID)(G). The identifying codes of the direct product G(1) x G(2), where G(1) is a complete graph and G(2) is a complete/regular/complete bipartite graph, are known in the literature. In this paper, we find gamma(ID)(P-n x K-m) for n >= 3, and m >= 3 where P-n is a path of length n, and K-m is a complete graph on m vertices.