Parameterized vertex deletion problems for hereditary graph classes with a block property

TL;DR

This paper introduces parameterized algorithms and kernelization results for a graph modification problem that aims to limit block sizes and properties within hereditary graph classes, providing efficient solutions under various conditions.

Contribution

It establishes fixed-parameter algorithms and kernel bounds for the Bounded P-Block Vertex Deletion problem across different hereditary graph classes.

Findings

Algorithm with runtime $2^{O(k ext{log} d)}n^{O(1)}$ for classes with polynomial recognition.

Optimality of the algorithm when $ ext{P}$ contains all split graphs unless ETH fails.

Existence of a kernel with $O(k^2 d^7)$ vertices.

Abstract

For a class of graphs $\mathcal{P}$, the Bounded $\mathcal{P}$-Block Vertex Deletion problem asks, given a graph $G$ on $n$ vertices and positive integers $k$ and $d$, whether there is a set $S$ of at most $k$ vertices such that each block of $G-S$ has at most $d$ vertices and is in $\mathcal{P}$. We show that when $\mathcal{P}$ satisfies a natural hereditary property and is recognizable in polynomial time, Bounded $\mathcal{P}$-Block Vertex Deletion can be solved in time $2^{O(k \log d)}n^{O(1)}$. When $\mathcal{P}$ contains all split graphs, we show that this running time is essentially optimal unless the Exponential Time Hypothesis fails. On the other hand, if $\mathcal{P}$ consists of only complete graphs, or only cycle graphs and $K_2$, then Bounded $\mathcal{P}$-Block Vertex Deletion admits a $c^{k}n^{O(1)}$-time algorithm for some constant $c$ independent of $d$. We also show that Bounded $\mathcal{P}$-Block Vertex Deletion admits a kernel with $O(k^2 d^7)$ vertices.