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Parameterized Vertex Integrity Revisited

Authors Tesshu Hanaka , Michael Lampis , Manolis Vasilakis , Kanae Yoshiwatari

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Author Details

Tesshu Hanaka
  • Department of Informatics, Kyushu University, Fukuoka, Japan
Michael Lampis
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France
Manolis Vasilakis
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France
Kanae Yoshiwatari
  • Department of Mathematical Informatics, Nagoya University, Aichi, Japan

Funding

This work is partially supported by ANR projects ANR-21-CE48-0022 (S-EX-AP-PE-AL) and ANR-18-CE40-0025-01 (ASSK), and JSPS KAKENHI Grant Numbers JP21K17707, JP21H05852, JP22H00513, and JP23H04388.

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Tesshu Hanaka, Michael Lampis, Manolis Vasilakis, and Kanae Yoshiwatari. Parameterized Vertex Integrity Revisited. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.58

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 very 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized Complexity
  • Treedepth
  • Vertex Integrity

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