Determination of the prime bound of a graph

Given a graph G, a subset M of V (G) is a module of G if for each v ∈ V (G) ∖M , v is adjacent to all the elements of M or adjacent to none of them. For instance, V (G), ∅ and {v} (v ∈ V (G)) are modules of G called trivial. Given a graph G, ωM(G) (respectively αM(G)) denotes the largest integer m such that there is a module M of G which is a clique (respectively a stable) set in G with ∣M ∣ = m. A graph G is prime if ∣V (G)∣ ≥ 4 and if all its modules are trivial. The prime bound of G is the smallest integer p(G) such that there is a prime graph H with V (H) ⊇ V (G), H[V (G)] = G and ∣V (H) ∖ V (G)∣ = p(G). We establish the following. For every graph G such that max(αM(G), ωM(G)) ≥ 2 and log2(max(αM(G), ωM(G))) is not an integer, p(G) = ⌈log2(max(αM(G), ωM(G)))⌉. Then, we prove that for every graph G such that max(αM(G), ωM(G)) = 2 k where k ≥ 1, p(G) = k or k+1. Moreover p(G) = k+1 if and only ifG or its complement admits exactly 2 isolated vertices. Lastly, we show that p(G) = 1 for every non prime graph G such that ∣V (G)∣ ≥ 4 and αM(G) = ωM(G) = 1.