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Structural Parameterizations of Vertex Integrity

Tatsuya Gima, Tesshu Hanaka, Yasuaki Kobayashi, Ryota Murai, Hirotaka Ono, Yota Otachi

TL;DR

A systematic study of structural parameterizations of the problem of computing the unweighted/weighted vertex integrity of the graph parameter vertex integrity, considering well-known parameters such as clique-width, treewidth, pathwidth, treedepth, modular-width, neighborhood diversity, twin cover number, and cluster vertex deletion number.

Abstract

The graph parameter vertex integrity measures how vulnerable a graph is to a removal of a small number of vertices. More precisely, a graph with small vertex integrity admits a small number of vertex removals to make the remaining connected components small. In this paper, we initiate a systematic study of structural parameterizations of the problem of computing the unweighted/weighted vertex integrity. As structural graph parameters, we consider well-known parameters such as clique-width, treewidth, pathwidth, treedepth, modular-width, neighborhood diversity, twin cover number, and cluster vertex deletion number. We show several positive and negative results and present sharp complexity contrasts. We also show that the vertex integrity can be approximated within an $\mathcal{O}(\log \mathsf{opt})$ factor.

Structural Parameterizations of Vertex Integrity

TL;DR

A systematic study of structural parameterizations of the problem of computing the unweighted/weighted vertex integrity of the graph parameter vertex integrity, considering well-known parameters such as clique-width, treewidth, pathwidth, treedepth, modular-width, neighborhood diversity, twin cover number, and cluster vertex deletion number.

Abstract

The graph parameter vertex integrity measures how vulnerable a graph is to a removal of a small number of vertices. More precisely, a graph with small vertex integrity admits a small number of vertex removals to make the remaining connected components small. In this paper, we initiate a systematic study of structural parameterizations of the problem of computing the unweighted/weighted vertex integrity. As structural graph parameters, we consider well-known parameters such as clique-width, treewidth, pathwidth, treedepth, modular-width, neighborhood diversity, twin cover number, and cluster vertex deletion number. We show several positive and negative results and present sharp complexity contrasts. We also show that the vertex integrity can be approximated within an factor.
Paper Structure (22 sections, 20 theorems, 19 equations, 6 figures)

This paper contains 22 sections, 20 theorems, 19 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. Overview of the proof ideas
  4. Positive results
  5. Negative results
  6. Related graph parameters
  7. Preliminaries
  8. Irredundant set
  9. Positive results
  10. The binary-weighted problem parameterized by $\mathop{\mathrm{\mathsf{nd}}}\nolimits$ or $\mathop{\mathrm{\mathsf{tc}}}\nolimits$
  11. The unary-weighted problem parameterized by $\mathop{\mathrm{\mathsf{mw}}}\nolimits$
  12. The unweighted problem parameterized by $\mathop{\mathrm{\mathsf{cvd}}}\nolimits$
  13. Reduction to nice instances of SubViCvd.
  14. Solving a nice instance of SubViCvd.
  15. The unary-weighted problem parameterized by $\mathop{\mathrm{\mathsf{cw}}}\nolimits$
  16. ...and 7 more sections

Key Result

Lemma 2.2

Let $G = (V,E,\mathop{\mathrm{\mathsf{w}}}\nolimits)$ be a vertex-weighted graph. If $S \subseteq V$ contains a redundant vertex $v$, then

Figures (6)

  • Figure 1: The complexity of Unweighted/Weighted Vertex Integrity with structural parameters. (See \ref{['sec:pre']} for the definitions of the acronyms.) The results without references are shown in this paper. The double, single, and rounded rectangles indicate paraNP-complete, W[1]-/W[2]-hard, and fixed-parameter tractable cases, respectively. A connection between two parameters implies that the one above generalizes the one below; that is, one below is lower-bounded by a function of the one above.
  • Figure 2: The complexity of Unweighted Vertex Integrity on graph classes. (See BrandstadtLS99 for the definitions of the graph classes.) The ones marked with asterisks are shown in this paper. The double and rounded rectangles indicate NP-complete and polynomial-time solvable cases, respectively. A connection between two graph classes indicates that the one above is a superclass of the one below.
  • Figure 3: The reduction in \ref{['thm:unary-vi-cvd-fvs_Wh']}.
  • Figure 4: The reduction in \ref{['thm:binary_subdivided_star_NPc']}.
  • Figure 5: The reduction in \ref{['thm:planar-bipartite']}. A size-$p$ vertex cover of $H$ is marked with circles.
  • ...and 1 more figures

Theorems & Definitions (38)

  • Lemma 2.2: DrangeDH16
  • proof
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Corollary 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • ...and 28 more