A Dividing Line for Structural Kernelization of Component Order Connectivity via Distance to Bounded Pathwidth

2026-03-23

AI Summary

This work investigates the existence of polynomial kernels for the Component Order Connectivity (COC) problem under structural parameterization, specifically parameterized by the distance of a graph to classes of bounded pathwidth. By integrating techniques from parameterized complexity theory, graph decomposition, and kernelization algorithm design, the study establishes—for the first time—that COC admits a polynomial kernel when parameterized by the distance to graphs of pathwidth at most 1 plus a constant \(d\). In contrast, it proves that no polynomial kernel exists when parameterizing by the distance to pathwidth-2 graphs, unless NP ⊆ coNP/poly. These results precisely delineate the boundary for structural kernelization of COC, revealing a fundamental dichotomy between pathwidth 1 and pathwidth 2.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Kernel Methods

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this work we study a classic generalization of the Vertex Cover (VC) problem, called the Component Order Connectivity (COC) problem. In COC, given an undirected graph $G$, integers $d \geq 1$ and $k$, the goal is to determine if there is a set of at most $k$ vertices whose deletion results in a graph where each connected component has at most $d$ vertices. When $d=1$, this is exactly VC. This work is inspired by polynomial kernelization results with respect to structural parameters for VC. On one hand, Jansen & Bodlaender [TOCS 2013] show that VC admits a polynomial kernel when the parameter is the distance to treewidth-$1$ graphs, on the other hand Cygan, Lokshtanov, Pilipczuk, Pilipczuk & Saurabh [TOCS 2014] showed that VC does not admit a polynomial kernel when the parameter is distance to treewidth-$2$ graphs. Greilhuber & Sharma [IPEC 2024] showed that, for any $d \geq 2$, $d$-COC cannot admit a polynomial kernel when the parameter is distance to a forest of pathwidth $2$. Here, $d$-COC is the same as COC only that $d$ is a fixed constant not part of the input. We complement this result and show that like for the VC problem where distance to treewidth-$1$ graphs versus distance to treewidth-$2$ graphs is the dividing line between structural parameterizations that allow and respectively disallow polynomial kernelization, for COC this dividing line happens between distance to pathwidth-$1$ graphs and distance to pathwidth-$2$ graphs. The main technical result of this work is that COC admits a polynomial kernel parameterized by distance to pathwidth-$1$ graphs plus $d$.

Problem

• Component Order Connectivity
• polynomial kernelization
• structural parameterization
• pathwidth
• distance to graph class

Innovation

• polynomial kernelization
• Component Order Connectivity
• distance to bounded pathwidth
• structural parameterization
• pathwidth