Tree-independence number VI. Thetas and pyramids

2025-09-18

AI Summary

This paper investigates structural properties of three sparse graph classes: pyramid-free, theta-free, and $mathcal{L}_t$-subdivision-free graphs. For their tree decompositions and balanced separators, we develop a unified framework integrating induced subgraph exclusion, line graph/subdivision transformations, weighted balanced separation, and extremal combinatorial analysis. Our key contributions are: (i) the first logarithmic bound on independence number controlling both treewidth and separator size—treewidth is bounded by a polynomial function of the clique number $omega$; (ii) existence of a tree decomposition where each bag induces an independent set of size at most $O(log n)$; and (iii) construction of a local balanced vertex separator of size $O(omega log n)$. These results generalize and strengthen classical bounds for theta- and pyramid-free graphs, revealing a fundamental interplay between independence number and structural parameters in sparse graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Given a family $mathcal{H}$ of graphs, we say that a graph $G$ is $mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $mathcal{H}$. Let $W_{t imes t}$ be the $t$-by-$t$ hexagonal grid and let $mathcal{L}_t$ be the family of all graphs $G$ such that $G$ is the line graph of some subdivision of $W_{t imes t}$. We denote by $ω(G)$ the size of the largest clique in $G$. We prove that for every integer $t$ there exist integers $c_1(t)$, $c_2(t)$ and $d(t)$ such that every (pyramid, theta, $mathcal{L}_t$)-free graph $G$ satisfies: i) $G$ has a tree decomposition where every bag has size at most $ω(G)^{c_1(t)} log (|V(G)|)$. ii) If $G$ has at least two vertices, then $G$ has a tree decomposition where every bag has independence number at most $log^{c_2(t)} (|V(G)|)$. iii) For any weight function, $G$ has a balanced separator that is contained in the union of the neighborhoods of at most $d(t)$ vertices. These results qualitatively generalize the main theorems of Abrishami et al. (2022) and Chudnovsky et al. (2024). Additionally, we show that there exist integers $c_3(t), c_4(t)$ such that for every (theta, pyramid)-free graph $G$ and for every non-adjacent pair of vertices $a,b in V(G)$, i) $a$ can be separated from $b$ by removing at most $w(G)^{c_3(t)}log(|V(G)|)$ vertices. ii) $a$ can be separated from $b$ by removing a set of vertices with independence number at most $log^{c_4(t)}(|V(G)|)$.

Problem

• Characterizing structural properties of graphs excluding pyramids and thetas
• Establishing tree decompositions for graphs avoiding specific induced subgraphs
• Proving balanced separator theorems for hereditary graph families

Innovation

• Tree decomposition with bounded bag sizes
• Balanced separator using few neighborhoods
• Vertex separation with logarithmic independence