Strongly connectable digraphs and non-transitive dice

Abstract

We give a new proof of the theorem of Boesch–Tindell and Farzad–Mahdian–Mahmoodian–Saberi–Sadri that a directed graph extends to a strongly connected digraph on the same vertex set if and only if it has no complete directed cut. Our proof bounds the number of edges needed for such an extension; we give examples to demonstrate sharpness. We apply the characterization to a problem on non-transitive dice.

Keywords

• strongly connectable digraph
• complete directed cut
• non-transitive dice