On the P3-hull number of Hamming graphs

Let S be a subset of vertices of a finite and simple graph G. If every vertex having two neighbors inside S is also in S, then S is P3-convex. The P3-convex hull H(S) of S is the smallest P3-convex set containing S. If H(S)=V(G), then S is a P3-hull set of G. The cardinality h(G) of a minimum P3-hull set in G is called the P3-hull number of G. In this paper, we prove that the P3-hull number of a Hamming graph Kr1□⋯□Krn which is the n-fold Cartesian product of complete graphs Kri, with ri>1 for 1≤i≤n, is equal to ⌈n∕2⌉+1.