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

Jakob Greilhuber, Roohani Sharma

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 $d$.

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

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 , integers and , the goal is to determine if there is a set of at most vertices whose deletion results in a graph where each connected component has at most vertices. When , 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- 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- graphs. Greilhuber & Sharma [IPEC 2024] showed that, for any , -COC cannot admit a polynomial kernel when the parameter is distance to a forest of pathwidth . Here, -COC is the same as COC only that 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- graphs versus distance to treewidth- 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- graphs and distance to pathwidth- graphs. The main technical result of this work is that COC admits a polynomial kernel parameterized by distance to pathwidth- graphs plus .
Paper Structure (10 sections, 34 theorems, 9 equations, 1 figure)

This paper contains 10 sections, 34 theorems, 9 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Further Results.
  3. Outlook.
  4. Technical Overview of Main Theorem 1
  5. Preliminiaries
  6. Reducing the Number of Connected Components
  7. Minimal Blocking-Sets of Caterpillars have Size at Most Two
  8. The Polynomial Kernel
  9. The Kernelization Algorithm
  10. The Essence of Caterpillars and Monoids

Key Result

Theorem 1

When $\mathcal{G}$ is the class of graphs with maximum degree at most $2$ the problem problem:coc_mod_to_g_plus_d admits a polynomial-kernel, and, for each $d \geq 1$, problem:dcoc_mod_to_g admits a polynomial kernel. When $\mathcal{G}$ is the class of planar graphs with maximum degree $3$ and $d$ a

Figures (1)

  • Figure 1: The replacement of a subcaterpillar $C$ with a caterpillar $C'$ that has the same essence $\gamma$, in the example $d = 2$. The solution-tight packings of $C$ and $C'$ are highlighted. The essence $\gamma$ fulfills $\gamma(0) = 0$, $\gamma(1) = 0$, $\gamma(2) = 1$, and $\gamma(3) = 3$. For any possible number $x$ of vertices of the caterpillar left of $C$ (or $C'$) which are in a connected component with the leftmost spine vertex of $C$ (or $C'$) we draw the corresponding solution within $C$ and $C'$ that also witnesses the values of the essence.

Theorems & Definitions (60)

  • Theorem 1
  • Conjecture 2
  • Lemma 2: Follows from Cockayne et al. cockayneMatchingsTransversalsHypergraphs1979
  • Definition 2: Solution-tight Packing
  • Definition 2: $\alpha$-merged Packing
  • Definition 2: Merged Graphs
  • Definition 2: Essence of a Caterpillar
  • Lemma 2
  • Definition 2: The Essence Monoid
  • Definition 2: Basic Functions
  • ...and 50 more