Domination in Diameter-Two Graphs and the 2-Club Cluster Vertex Deletion Parameter

TL;DR

This paper investigates domination problems on diameter-two graphs, focusing on their parameterized complexity relative to the 2-club cluster vertex deletion number, and explores sub-exponential algorithms and hardness results based on the ETH.

Contribution

It introduces the study of domination problems on 2-clubs with respect to the 2ccvd parameter and examines their complexity and sub-exponential solvability.

Findings

Identifies domination problems that are fixed-parameter tractable with respect to 2ccvd.

Provides hardness results for certain problems assuming the Exponential-Time Hypothesis.

Proposes a framework for problems solvable in sub-exponential time independent of specific input parameters.

Abstract

The s-club cluster vertex deletion number of a graph, or sccvd, is the minimum number of vertices whose deletion results in a disjoint union of s-clubs, or graphs whose diameter is bounded above by s. We launch a study of several domination problems on diameter-two graphs, or 2-clubs, and study their parameterized complexity with respect to the 2ccvd number as main parameter. We further propose to explore the class of problems that become solvable in sub-exponential time when the running time is independent of some input parameter. Hardness of problems for this class depends on the Exponential-Time Hypothesis. We give examples of problems that are in the proposed class and problems that are hard for it.