Two-Path Convexity and Bipartite Tournaments of Small Rank

We study two-path convexity in bipartite tournaments. For a bipartite tour-nament, we obtain both a necessary condition and a sufficient condition on the adjacency matrix for its rank to be two. We then investigate 4-cycles in bipartite tournaments of small rank. We show that every vertex in a bipartite tournament of rank two lies on a four cycle, and bipartite tournaments with a maximum number of 4-cycles do not necessarily have minimum rank.