Clique-cutsets beyond chordal graphs

TL;DR

This paper explores the structure of graphs excluding certain Truemper configurations, analyzing their properties and complexities, and providing polynomial bounds for coloring these classes.

Contribution

It characterizes graphs with limited Truemper configurations and studies their recognition, clique, stable set, and coloring problems, offering new structural insights.

Findings

Polynomial algorithms for recognition and coloring

Structural characterization of graphs with restricted Truemper configurations

Polynomial chi-bounding functions for these classes

Abstract

Truemper configurations (thetas, pyramids, prisms, and wheels) have played an important role in the study of complex hereditary graph classes (e.g. the class of perfect graphs and the class of even-hole-free graphs), appearing both as excluded configurations, and as configurations around which graphs can be decomposed. In this paper, we study the structure of graphs that contain (as induced subgraphs) no Truemper configurations other than (possibly) universal wheels and twin wheels. We also study several subclasses of this class. We use our structural results to analyze the complexity of the recognition, maximum weight clique, maximum weight stable set, and optimal vertex coloring problems for these classes. Furthermore, we obtain polynomial chi-bounding functions for these classes.