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Feedback Vertex Set for pseudo-disk graphs in subexponential FPT time

Gaétan Berthe, Marin Bougeret, Daniel Gonçalves, Jean-Florent Raymond

Abstract

In this paper, we investigate the existence of parameterized algorithms running in subexponential time for two fundamental cycle-hitting problems: Feedback Vertex Set (FVS) and Triangle Hitting (TH). We focus on the class of pseudo-disk graphs, which forms a common generalization of several graph classes where such results exist, like disk graphs and square graphs. In these graphs, we show that TH can be solved in time $2^{O(k^{3/4}\log k)}n^{O(1)}$, and given a geometric representation FVS can be solved in time $2^{O(k^{6/7}\log k)}n^{O(1)}$.

Feedback Vertex Set for pseudo-disk graphs in subexponential FPT time

Abstract

In this paper, we investigate the existence of parameterized algorithms running in subexponential time for two fundamental cycle-hitting problems: Feedback Vertex Set (FVS) and Triangle Hitting (TH). We focus on the class of pseudo-disk graphs, which forms a common generalization of several graph classes where such results exist, like disk graphs and square graphs. In these graphs, we show that TH can be solved in time , and given a geometric representation FVS can be solved in time .

Paper Structure

This paper contains 13 sections, 8 theorems, 1 equation, 4 figures.

Table of Contents

  1. Introduction
  2. Context.
  3. Contribution and techniques.
  4. Organization of the paper.
  5. Preliminaries
  6. Basics
  7. Graph classes
  8. Pseudo-disk graphs.
  9. Representation of pseudo-disk graphs.
  10. Properties of pseudo-disk graphs
  11. Preliminary branching step
  12. Hitting cycles in pseudo-disk graphs
  13. Proof of \ref{['lemma:red-fvs-pseudo']}

Key Result

lemma 1

Given a pseudo-disk graph $G$, deleting any vertex, or any edge $uv$ that does not belong to a triangle, or contracting any edge $uv$ that does not belong to a triangle results in a pseudo-disk graph. Furthermore, given a representation of $G$ of ply $p$, one can update the representation in linear

Figures (4)

  • Figure 1: Three forbidden intersections between pseudo-disks, and two avoidable examples.
  • Figure 2: A pseudo-disk representation ${\overrightarrow{P_\mathcal{S}}}$ and its dual ${\overrightarrow{P^*_\mathcal{S}}}$.
  • Figure 3: Left: From $M$ to $M'$. The pseudo-disks of $M$ are filled. Among the others, those added to $M'$ have solid border. Right: Here, the filled pseudo-disks belong to $M'$. In dashed green, the inner pseudo-disks of $I_{M'}$. In dashed red, the border pseudo-disks of $B_{M'}$. In dashed orange, the only pseudo-disk $v \in O_{M'}$. For any pseudo-disk $u$ not in $M'$, we depicted its hosted graph $H_u$, which is defined in \ref{['def:Ht-intro']}.
  • Figure 4: Proof of Property \ref{['e:ee']} (left) The region ${\mathcal{R}}^*$. (right) The pseudo-disks $\mathcal{D}_u$ and $\mathcal{D}_{v_2}$.

Theorems & Definitions (13)

  • lemma 1
  • proof
  • theorem 1: lee2016separators and DVORAK2019137
  • lemma 2: See the proof of Lemma 16 in lokshtanov2024linearedges
  • corollary 1
  • lemma 3
  • proof
  • corollary 2: Corollary 10 in berthe2023subexponential
  • theorem 2
  • lemma 4
  • ...and 3 more