Transitive orientations in bull-reducible Berge graphs

TL;DR

This paper proves that bull-reducible Berge graphs without antiholes are either weakly chordal, have a homogeneous set, or are transitively orientable, leading to an efficient coloring algorithm.

Contribution

It establishes a structural characterization of bull-reducible Berge graphs without antiholes and provides a polynomial-time coloring algorithm for them.

Findings

Bull-reducible Berge graphs without antiholes are weakly chordal or have a homogeneous set or are transitively orientable.

The paper introduces a polynomial-time algorithm for coloring such graphs.

Structural properties of these graphs facilitate efficient algorithms.

Abstract

A bull is a graph with five vertices $r, y, x, z, s$ and five edges $ry$, $yx$, $yz$, $xz$, $zs$. A graph $G$ is bull-reducible if no vertex of $G$ lies in two bulls. We prove that every bull-reducible Berge graph $G$ that contains no antihole is weakly chordal, or has a homogeneous set, or is transitively orientable. This yields a fast polynomial time algorithm to color exactly the vertices of such a graph.