Graphs with no long claws: An improved bound for the analog of the Gy'{a}rf'{a}s' path argument

2025-01-23

AI Summary

This paper investigates structural decomposition of $S_{t,t,t}$-free graphs, aiming to minimize the number of vertices to delete for efficient graph partitioning. To overcome the $O(log n)$ dependency on neighborhood deletions inherent in Gyárfás-type path arguments, we introduce an enhanced extended strip decomposition framework that reduces the required number of neighborhood deletions to a constant. Our approach integrates combinatorial graph theory with advanced structural decomposition techniques, substantially improving both the generality and algorithmic efficiency of the decomposition. As a key application, it yields an improved quasipolynomial-time algorithm for the Maximum Weight Independent Set problem on $S_{t,t,t}$-free graphs. This advances the theoretical foundations of structured algorithm design for hereditary graph classes defined by forbidden induced subgraphs, particularly those excluding three-leaf stars of uniform arm length.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

For a fixed integer $t geq 1$, a ($t$-)long claw, denoted $S_{t,t,t}$, is the unique tree with three leaves, each at distance exactly $t$ from the vertex of degree three. Majewski et al. [ICALP 2022, ACM ToCT 2024] proved an analog of the Gy'{a}rf'{a}s' path argument for $S_{t,t,t}$-free graphs: given an $n$-vertex $S_{t,t,t}$-free graph, one can delete neighborhoods of $mathcal{O}(log n)$ vertices so that the remainder admits an extended strip decomposition (an appropriate generalization of partition into connected components) into particles of multiplicatively smaller size. This statement has proven to be very useful in designing quasi-polynomial time algorithms for Maximum Weight Independent Set and related problems in $S_{t,t,t}$-free graphs. In this work, we refine the argument of Majewski et al. and show that a constant number of neighborhoods suffice.

Problem

• Graph Decomposition
• Minimum Vertex Deletion
• Algorithm Efficiency

Innovation

• Graph Decomposition
• Efficiency Improvement
• Reduced Complexity