Parameterized vertex deletion problems for hereditary graph classes with a block property
\'Edouard Bonnet, Nick Brettell, O-joung Kwon, D\'aniel Marx

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 , the Bounded -Block Vertex Deletion problem asks, given a graph on vertices and positive integers and , whether there is a set of at most vertices such that each block of has at most vertices and is in . We show that when satisfies a natural hereditary property and is recognizable in polynomial time, Bounded -Block Vertex Deletion can be solved in time . When contains all split graphs, we show that this running time is essentially optimal unless the Exponential Time Hypothesis fails. On the other hand, if consists of only complete graphs, or only cycle graphs and , then Bounded -Block Vertex Deletion admits a -time algorithm for some constant independent of . We also show…
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Taxonomy
TopicsAdvanced Graph Theory Research · Genome Rearrangement Algorithms
