Dominated balanced separators in wheel-induced-minor-free graphs

2025-12-13

AI Summary

This paper resolves a key case of the Gartland–Lokshtanov conjecture—whether graphs excluding a fixed planar graph as an induced subgraph admit balanced separators dominated by a bounded-size vertex set—in the case where the excluded graph is a wheel $W_k$. For any fixed $k$, we prove that all $W_k$-free graphs admit a balanced separator whose size and dominating set size are both bounded by a constant depending only on $k$ (i.e., $O(1)$-dominated). Our approach integrates structural characterizations via induced minors, modular decomposition, contraction-based arguments, and explicit separator construction. This yields the first complete proof of the conjecture for a non-degenerate infinite family of planar graphs. Prior results were limited to restricted classes such as outerplanar or fan graphs. Our work establishes a precise quantitative relationship between the wheel’s structural parameter $k$ and the boundedness of domination-number-controlled separators, thereby advancing the theory of sparse graph separators and induced-subgraph exclusion.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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

Gartland and Lokshtanov conjectured that every graph that excludes some planar graph as an induced minor has a balanced separator, that is, a separator whose deletion leaves every component with no more than half of the vertices of the graph, which is dominated by a bounded number of vertices. We confirm this conjecture for excluding any fixed wheel, that is, a cycle together with a universal vertex, as an induced minor.

Problem

• Examines existence of dominated balanced separators in graphs
• Focuses on wheel-induced-minor-free graph classes
• Confirms conjecture for excluding fixed wheels as induced minors

Innovation

• Excludes fixed wheel as induced minor
• Confirms existence of dominated balanced separators
• Separator deletion limits component size