Polynomial-time recognition and maximum independent set in Burling graphs

TL;DR

This work settles the computational aspects of Burling graphs by giving a polynomial-time recognition algorithm and a constructive method to obtain strict frame representations via Burling sets. Once such a representation is available, the authors adapt Gavril’s dynamic programming framework to compute maximum independent sets (weighted) in Burling graphs in polynomial time, despite Burling graphs not being $\chi$-bounded. The results connect geometric frame representations with combinatorial Burling structures, enabling efficient MIS and offering a negative answer to whether all polynomial-time MIS classes are $\\chi$-bounded. Overall, Burling graphs are shown to be a hereditary class supporting efficient MIS without $\\chi$-boundedness, advancing the understanding of structural graph theory at the intersection of geometry and algorithms.

Abstract

A Burling graph is an induced subgraph of some graph in Burling's construction of triangle-free high-chromatic graphs. We provide a polynomial-time algorithm which decides whether a given graph is a Burling graph and if it is, constructs its intersection model by rectangular frames. That model 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 $χ$-bounded.

Figures (5)

• Figure 1: Strict family of frames: (a) required configuration of every intersecting pair of frames; (b)--(d) disallowed configurations.
• Figure 2: Correspondence between the relations $\prec$ and $\curvearrowright$ in a Burling set and configurations of pairs of frames in its strict frame representation.
• Figure 3: Distinguished elements of a Burling set in a frame representation: $a$ is a root, $b$ and $f$ are probes, and all three are exposed; $c$ and $d$ are exposed but neither roots not probes; $e$ is not exposed.
• Figure 4: The second condition in the algorithm for $\mathfrak{B}(X,S)$ with $C_1$ and $C_2$ the components of $S-\{r\}$.
• Figure 5: A solution to a subproblem $\mathfrak{R}(X,r,S)$. The graph induced by $S-N(r)$ has six components $C_1,\ldots,C_6$, of which $C_1$ and $C_2$ are inner while $C_3,\ldots,C_6$ are outer. Gray areas are where particular components are represented.

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Theorem 3: PT23
• Lemma 4: PT23
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7