Some Remarks on the Convexity Number of a Graph

Given a subgraph H in a graph G, we say H is convex if given any two vertices u and v in H, then H contains each shortest u–v path in G. As defined in [5], [8] and other places the convexity number of G, denoted con(G), is the order of a largest convex proper subgraph of G. We note that if G is not a complete graph, then the clique number of G forms a lower bound on con(G). As we shall see, in many graphs the clique number and convexity number are equal. We exploit this equality to show that computation of con is NP-complete, provide a Nordhaus-Gaddum type bound for con and produce a Ramsey type theorem for con. We shall see the latter two problems are closely related to classical Ramsey theory. In doing so, we answer a problem posed by Chartrand and Zhang in [5].