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Parameterized Vertex Integrity Revisited

Tesshu Hanaka, Michael Lampis, Manolis Vasilakis, Kanae Yoshiwatari

TL;DR

This work analyzes the parameterized complexity of computing vertex integrity, introducing both unweighted and weighted variants. It establishes a clear boundary of tractability across width parameters: Vertex Integrity is W[1]-hard for treedepth and for fes+Δ, yet FPT by max-leaf number, and also FPT for weighted Vertex Integrity when parameterized by modular width or vertex cover, with the latter offering a single-exponential dependence. The results extend existing work to binary-weighted scenarios for modular width and provide a refined understanding of which structural parameters enable efficient algorithms. These findings have implications for leveraging vertex integrity in designing fixed-parameter algorithms for hard graph problems and guide directions for potential FPT approximations or hardness under other width measures.

Abstract

Vertex integrity is a graph parameter that measures the connectivity of a graph. Informally, its meaning is that a graph has small vertex integrity if it has a small separator whose removal disconnects the graph into connected components which are themselves also small. Graphs with low vertex integrity are extremely structured; this renders many hard problems tractable and has recently attracted interest in this notion from the parameterized complexity community. In this paper we revisit the NP-complete problem of computing the vertex integrity of a given graph from the point of view of structural parameterizations. We present a number of new results, which also answer some recently posed open questions from the literature. Specifically: We show that unweighted vertex integrity is W[1]-hard parameterized by treedepth; we show that the problem remains W[1]-hard if we parameterize by feedback edge set size (via a reduction from a Bin Packing variant which may be of independent interest); and complementing this we show that the problem is FPT by max-leaf number. Furthermore, for weighted vertex integrity, we show that the problem admits a single-exponential FPT algorithm parameterized by vertex cover or by modular width, the latter result improving upon a previous algorithm which required weights to be polynomially bounded.

Parameterized Vertex Integrity Revisited

TL;DR

This work analyzes the parameterized complexity of computing vertex integrity, introducing both unweighted and weighted variants. It establishes a clear boundary of tractability across width parameters: Vertex Integrity is W[1]-hard for treedepth and for fes+Δ, yet FPT by max-leaf number, and also FPT for weighted Vertex Integrity when parameterized by modular width or vertex cover, with the latter offering a single-exponential dependence. The results extend existing work to binary-weighted scenarios for modular width and provide a refined understanding of which structural parameters enable efficient algorithms. These findings have implications for leveraging vertex integrity in designing fixed-parameter algorithms for hard graph problems and guide directions for potential FPT approximations or hardness under other width measures.

Abstract

Vertex integrity is a graph parameter that measures the connectivity of a graph. Informally, its meaning is that a graph has small vertex integrity if it has a small separator whose removal disconnects the graph into connected components which are themselves also small. Graphs with low vertex integrity are extremely structured; this renders many hard problems tractable and has recently attracted interest in this notion from the parameterized complexity community. In this paper we revisit the NP-complete problem of computing the vertex integrity of a given graph from the point of view of structural parameterizations. We present a number of new results, which also answer some recently posed open questions from the literature. Specifically: We show that unweighted vertex integrity is W[1]-hard parameterized by treedepth; we show that the problem remains W[1]-hard if we parameterize by feedback edge set size (via a reduction from a Bin Packing variant which may be of independent interest); and complementing this we show that the problem is FPT by max-leaf number. Furthermore, for weighted vertex integrity, we show that the problem admits a single-exponential FPT algorithm parameterized by vertex cover or by modular width, the latter result improving upon a previous algorithm which required weights to be polynomially bounded.
Paper Structure (12 sections, 5 theorems, 1 equation, 1 figure)

This paper contains 12 sections, 5 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vertex Integrity
  4. Graph Parameters
  5. Treedepth
  6. Feedback Edge Set plus Maximum Degree
  7. Preliminary Tools
  8. Hardness Result
  9. Max-Leaf Number
  10. Modular Width
  11. Vertex Cover Number
  12. Conclusion

Key Result

Proposition 1

A graph with a $\text{\rm wvi}(k)$-set has an irredundant $\text{\rm wvi}(k)$-set.

Figures (1)

  • Figure 1: The parameterized complexity of Unweighted Vertex Integrity, with the underlined parameters indicating our results. A connection between two parameters implies that the one above generalizes the one below; that is, the one below is lower-bounded by a function of the one above. All of our FPT algorithms have single-exponential parametric dependence, while the ones for $\text{\rm vc}$ and $\text{\rm mw}$ extend to the weighted case as well.

Theorems & Definitions (5)

  • Proposition 1: DrangeDH16Gima2023
  • Theorem 2
  • Theorem 8
  • Theorem 9
  • Theorem 11