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 ∈ V, the set N[v] is the vertex v and all its neighbors. A subset D ⊆ V (G) is a dominating set of G if for every v ∈ V (G), N[v] ∩ D ≠ ∅. And a subset F ⊆ V (G) is a separating set of G if for every distinct pair u, v ∈ V (G), N[u] ∩ F ≠ N[v] ∩ F. An identifying code of G is a subset C ⊆ V (G) that is dominating as well as separating. The minimum cardinality of an identifying code in a graph G is denoted by γID(G). The identifying codes of the direct product G1 × G2, where G1 is a complete graph and G2 is a complete/regular/complete bipartite graph, are known in the literature. In this paper, we find γID(Pn × Km) for n ≥ 3, and m ≥ 3 where Pn is a path of length n, and Km is a complete graph on m vertices.