Geodesic Convexity and Cartesian Products in Graphs

In this work we investigate the behavior of various geodesic convexity parameters with respect to the Cartesian product operation for graphs. First, we show that the convex sets arising from geodesic convexity in a Cartesian product of graphs are exactly the same as the convex sets arising from the usual binary operation ⊕ for making a convexity space out of the Cartesian product of any two convexity spaces. To be more precise, we prove that if (G1, C1), (G2, C2) are two graph (geodesic) convexity spaces and if (G1×G2, C) is the graph (geodesic) convexity space determined by the graph G1 ×G2, then C = C1 ⊕ C2 = {A×B | A ∈ C1, B ∈ C2}. Second, we study results involving a number of classical and graph-theoretic convexity parameters as applied to Cartesian products of graphs. For example, concerning geodetic numbers of graphs, we prove that for every two nontrivial graphs G, H such that gn(G) = p ≥ gn(H) = q ≥ 1, p ≤ gn(G×H) ≤ pq − q, and that both bounds are tight.