Polynomial-time recognition and maximum independent set in Burling graphs

TL;DR

This paper presents a polynomial-time algorithm to recognize Burling graphs, construct their strict frame representations, and find maximum independent sets, establishing these graphs as a unique hereditary class with such properties.

Contribution

It introduces the first polynomial-time recognition and maximum independent set algorithms for Burling graphs, a class not known to be χ-bounded.

Findings

Polynomial-time recognition algorithm for Burling graphs

Construction of strict frame representations in polynomial time

Polynomial-time maximum independent set algorithm for Burling graphs

Abstract

A Burling graph is an induced subgraph of some graph in Burling's construction of triangle-free high-chromatic graphs. Equivalently, a Burling graph is a graph that admits a so-called strict frame representation. We provide a polynomial-time algorithm to decide whether a given graph is a Burling graph and if it is, to construct its strict frame representation. The representation then enables a polynomial-time algorithm for the maximum independent set problem in Burling graphs. As a consequence, we establish Burling graphs as the first known hereditary class of graphs that admits such an algorithm while not being $\chi$-bounded.