Kick the cliques

TL;DR

This paper presents a subexponential fixed-parameter tractable algorithm for the $K_r$-Cover problem applicable to various graph classes, improving efficiency in destroying all $r$-cliques with limited vertex removal.

Contribution

The paper introduces a novel FPT algorithm for $K_r$-Cover that operates efficiently on graph classes with specific clique and treewidth properties, extending applicability to pseudo-disk, string, and minor-free graphs.

Findings

Algorithm runs in subexponential FPT time for specific graph classes.

Effective in pseudo-disk, string, and $H$-minor-free graphs.

Provides bounds depending on parameters $k$, $r$, and graph class properties.

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.