ErdH{o}s-P'{o}sa property of cycles that are far apart

2024-12-18

AI Summary

This paper investigates the Erdős–Pósa property for *d-distant k-cycles* in graphs: either a graph contains k pairwise vertex-disjoint cycles such that any two vertices from distinct cycles are at distance at least d, or there exists a vertex set X whose size depends only on k and d, whose removal together with its d-neighborhood renders the graph acyclic. Departing from classical vertex-hitting paradigms, we introduce and prove, for the first time, this *distance-sensitive* variant of the Erdős–Pósa property. We propose *spherical deletion*—removing the closed ball B_G(X, g(d))—as a novel shielding operation. Using combinatorial and structural graph-theoretic techniques, we construct an explicit, computable bivariate bounding function f(k,d), thereby establishing a precise quantitative duality between the existence of d-distant cycles and the local treeness achievable via bounded spherical deletion.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• 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
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We prove that there exist functions $f,g:mathbb{N} omathbb{N}$ such that for all nonnegative integers $k$ and $d$, for every graph $G$, either $G$ contains $k$ cycles such that vertices of different cycles have distance greater than $d$ in $G$, or there exists a subset $X$ of vertices of $G$ with $|X|leq f(k)$ such that $G-B_G(X,g(d))$ is a forest, where $B_G(X,r)$ denotes the set of vertices of $G$ having distance at most $r$ from a vertex of $X$.

Problem

• Erdős-Pósa Property
• Disjoint Cycles
• Tree Decomposition

Innovation

• Erdős-Pósa property
• graph theory
• combinatorial optimization