Strongly connectable digraphs and non-transitive dice

TL;DR

This paper provides a new proof for a theorem characterizing when a directed graph can be extended to a strongly connected digraph without complete directed cuts, and applies this to non-transitive dice.

Contribution

It introduces a new proof with bounds on the number of edges needed for extension and applies the result to non-transitive dice problems.

Findings

Bound on edges needed for strongly connected extension

Examples demonstrating sharpness of bounds

Application to 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.