Complementation in t-perfect graphs

TL;DR

This paper investigates the properties of t-perfect graphs, focusing on their behavior under complementation, and identifies specific pairs of graphs that are minimally t-imperfect along with their complements.

Contribution

It provides new insights into t-perfection under complementation and identifies five pairs of graphs with both graphs and their complements minimally t-imperfect.

Findings

Only five pairs of graphs have both the graphs and their complements minimally t-imperfect.

t-perfect graphs are not closed under complementation.

The work advances understanding of t-perfection in relation to graph complementation.

Abstract

Inspired by applications of perfect graphs in combinatorial optimization, Chv\'{a}tal defined t-perfect graphs in 1970s. The long efforts of characterizing t-perfect graphs started immediately, but embarrassingly, even a working conjecture on it is still missing after nearly 50 years. Unlike perfect graphs, t-perfect graphs are not closed under substitution or complementation. A full characterization of t-perfection with respect to substitution has been obtained by Benchetrit in his Ph.D. thesis. Through the present work we attempt to understand t-perfection with respect to complementation. In particular, we show that there are only five pairs of graphs such that both the graphs and their complements are minimally t-imperfect.