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Kick the cliques

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

TL;DR

The paper develops a generic subexponential FPT framework for the $K_r$-Cover problem on hereditary graph classes defined by two structural properties relating clique counts and treewidth. It combines a large-clique pruning phase with a petals-based branching strategy that expands a small core $M'$ while ensuring the remainder of the graph becomes irrelevant or has bounded treewidth, enabling dynamic programming. The main theorem shows subexponential time $2^{O_{r,oldsymbol{\phi}}(k^{oldsymbol{\varepsilon}} ext{log}\,k)}$ for classes with property $P_r(oldsymbol{\phi},oldsymbol{\\gamma},oldsymbol{\\alpha})$, with explicit instantiations yielding concrete bounds for pseudo-disk graphs, map graphs, $K_{t,t}$-free string graphs, and $H$-minor-free graphs. This significantly broadens subexponential parameterized algorithm techniques beyond bidimensional settings, applying to geometric/intersection graphs and minor-closed classes through principled clique-control and treewidth-based DP. The results also illuminate why large bicliques hinder string graphs and outline a program to characterize graph classes supporting such subexponential algorithms.

Abstract

In the $K_r$-Cover problem, given a graph $G$ and an integer $k$ one has to decide if there exists a set of at most $k$ vertices whose removal destroys all $r$-cliques of $G$. In this paper we give an algorithm for $K_r$-Cover that runs in subexponential FPT time on graph classes satisfying two simple conditions related to cliques and treewidth. As an application we show that our algorithm solves $K_r$-Cover in time * $2^{O_r\left (k^{(r+1)/(r+2)}\log k \right)} \cdot n^{O_r(1)}$ in pseudo-disk graphs and map-graphs; * $2^{O_{t,r}(k^{2/3}\log k)} \cdot n^{O_r(1)}$ in $K_{t,t}$-subgraph-free string graphs; and * $2^{O_{H,r}(k^{2/3}\log k)} \cdot n^{O_r(1)}$ in $H$-minor-free graphs.

Kick the cliques

TL;DR

The paper develops a generic subexponential FPT framework for the -Cover problem on hereditary graph classes defined by two structural properties relating clique counts and treewidth. It combines a large-clique pruning phase with a petals-based branching strategy that expands a small core while ensuring the remainder of the graph becomes irrelevant or has bounded treewidth, enabling dynamic programming. The main theorem shows subexponential time for classes with property , with explicit instantiations yielding concrete bounds for pseudo-disk graphs, map graphs, -free string graphs, and -minor-free graphs. This significantly broadens subexponential parameterized algorithm techniques beyond bidimensional settings, applying to geometric/intersection graphs and minor-closed classes through principled clique-control and treewidth-based DP. The results also illuminate why large bicliques hinder string graphs and outline a program to characterize graph classes supporting such subexponential algorithms.

Abstract

In the -Cover problem, given a graph and an integer one has to decide if there exists a set of at most vertices whose removal destroys all -cliques of . In this paper we give an algorithm for -Cover that runs in subexponential FPT time on graph classes satisfying two simple conditions related to cliques and treewidth. As an application we show that our algorithm solves -Cover in time * in pseudo-disk graphs and map-graphs; * in -subgraph-free string graphs; and * in -minor-free graphs.
Paper Structure (10 sections, 32 theorems, 17 equations, 1 figure)

This paper contains 10 sections, 32 theorems, 17 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dealing with large cliques
  4. Picking petals
  5. Kick the cliques
  6. Applications
  7. Pseudo-disk graphs and map graphs
  8. String graphs
  9. Minor-closed classes
  10. Open problems

Key Result

Theorem 1.1

For every apexA graph is apex if the deletion of some vertex yields a planar graph. graph $H$ and every $r\in \mathbb{N}$ there is an algorithm solving $K_r$-CoverActually the statement also applies to $F$-Cover for any $F$. in $H$-minor-free graphs in time $2^{O_{H,r}\left (\sqrt{k} \right )} \cdot

Figures (1)

  • Figure 1: A $3$-clique $X$ with two $5$-petals that live in $G-M$. Here $\mathcal{D}$ contains two hyperedges, represented in orange. Observe that no hyperedge of $\mathcal{D}$ is contained in $X$, so $X$ is a lush $3$-clique.

Theorems & Definitions (52)

  • Theorem 1.1: from fomin2011bidimensionality
  • Theorem 1.2
  • Theorem 1.3: berthe24
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 3.1
  • ...and 42 more