A single-exponential fixed-parameter algorithm for Distance-Hereditary Vertex Deletion

TL;DR

This paper introduces the first single-exponential fixed-parameter algorithm for deleting vertices to obtain distance-hereditary graphs, advancing the understanding of vertex deletion problems and their computational complexity.

Contribution

It presents the first single-exponential fixed-parameter tractable algorithm for vertex deletion to distance-hereditary graphs, with matching lower bounds based on the exponential time hypothesis.

Findings

Algorithm is fixed-parameter tractable with single-exponential complexity.

Matching lower bounds confirm the optimality of the algorithm.

Application to NP-hard problems using the deletion set as a parameter.

Abstract

Vertex deletion problems ask whether it is possible to delete at most $k$ vertices from a graph so that the resulting graph belongs to a specified graph class. Over the past years, the parameterized complexity of vertex deletion to a plethora of graph classes has been systematically researched. Here we present the first single-exponential fixed-parameter tractable algorithm for vertex deletion to distance-hereditary graphs, a well-studied graph class which is particularly important in the context of vertex deletion due to its connection to the graph parameter rank-width. We complement our result with matching asymptotic lower bounds based on the exponential time hypothesis. As an application of our algorithm, we show that a vertex deletion set to distance-hereditary graphs can be used as a parameter which allows single-exponential fixed-parameter tractable algorithms for classical NP-hard problems.