Interval Tournaments A directed graph is an interval digraph if to each vertex v there corresponds an ordered pair of intervals (S(v),T(v)) such that u beats v if and only if the intersection of S(u) and T(v) is nonempty. A tournament is an oriented complete graph. We characterize the tournaments that are interval digraphs via the existence of a large transitive subtournament and by forbidden subtournaments. A bipartite graph is an interval bigraph if to each vertex there corresponds an interval such that vertices are adjacent if and only if their corresponding intervals intersect and each vertex belongs to a different partite set. We capitalize on the equivalence of the models for interval digraphs and interval bigraphs and use results of Das, Roy, Sen, and West for interval digraphs, and results of Muller for interval bigraphs. We will also comment on some related open questions.

This is joint work with Dave Brown and Art Busch.