Orientations of graphs with at most one directed path between every pair of vertices

TL;DR

This paper investigates graphs that can be oriented so that there is at most one directed path between any two vertices, exploring their properties, computational complexity, and providing algorithms for specific cases.

Contribution

It introduces a new graph family with small independence number that admits KT orientations and establishes NP-completeness results for deciding KT orientations.

Findings

Construction of a graph family with small independence number admitting KT orientations

NP-completeness of deciding KT orientations for general graphs and planar graphs

Efficient algorithm for graphs with maximum degree 3

Abstract

Given a graph $G$, we say that an orientation $D$ of $G$ is a KT orientation if, for all $u, v \in V(D)$, there is at most one directed path (in any direction) between $u$ and $v$. Graphs that admit such orientations have been used by Kierstead and Trotter (1992), Carbonero, Hompe, Moore, and Spirkl (2023), Bria\'nski, Davies, and Walczak (2024), and Gir\~ao, Illingworth, Powierski, Savery, Scott, Tamitegami, and Tan (2024) to construct graphs with large chromatic number and small clique number that served as counterexamples to various conjectures. Motivated by this, we consider which graphs admit KT orientations (named after Kierstead and Trotter). In particular, we construct a graph family with small independence number (sublinear in the number of vertices) which admits a KT orientation. We show that the problem of determining whether a given graph admits a KT orientation is NP-complete, even if we restrict ourselves to planar graphs. Finally, we provide an algorithm to decide if a graph with maximum degree at most 3 admits a KT orientation, whereas, for graphs with maximum degree 4, the problem remains NP-complete.