Let diam min (G) denote the minimum diameter of a strong orientation of G and let G H denote the strong product of graphs G and H. In this paper we prove that diam min (P m P n) = diam(P m P n) for m, n ≥ 5, m = n, and diam min (P m P n) = diam(P m P n) + 1 for m, n ≥ 5, m = n. We also prove that diam min (G H) ≤ max {diam min (G), diam min (H)} for any connected bridgeless graphs G and H.
A set S ⊆ V (G) is a vertex k-cut in a graph G = (V (G), E(G)) if G − S has at least k connected components. The k-connectivity of G, denoted as κ k (G), is the minimum cardinality of a vertex k-cut in G. We give several constructions of a set S such that (G2H) − S has at least three connected components. Then we prove that for any 2-connected graphs G and H, of order at least six, one of the defined sets S is a minimum vertex 3-cut in G2H. This yields a formula for κ 3 (G2H).
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