On Fork-free T-perfect Graphs

TL;DR

This paper advances understanding of fork-free t-perfect graphs by characterizing their properties, proving they are strongly t-perfect and three-colorable, and providing polynomial-time algorithms for recognition and coloring.

Contribution

It offers a new characterization of fork-free t-perfect graphs and develops efficient algorithms for their recognition and coloring.

Findings

Fork-free t-perfect graphs are strongly t-perfect.

They are three-colorable.

Polynomial-time algorithms exist for recognition and coloring.

Abstract

In an attempt to understanding the complexity of the independent set problem, Chv{\'a}tal defined t-perfect graphs. While a full characterization of this class is still at large, progress has been achieved for claw-free graphs [Bruhn and Stein, Math.\ Program.\ 2012] and $P_{5}$-free graphs [Bruhn and Fuchs, SIAM J.\ Discrete Math.\ 2017]. We take one more step to characterize fork-free t-perfect graphs, and show that they are strongly t-perfect and three-colorable. We also present polynomial-time algorithms for recognizing and coloring these graphs.