Parameterized Complexity of Dominating Set Variants in Almost Cluster and Split Graphs

TL;DR

This paper studies the parameterized complexity of dominating set variants in graphs close to cluster and split graphs, providing improved algorithms and lower bounds, and exploring kernelization limits.

Contribution

It introduces new fixed-parameter tractable algorithms for dominating set variants in graphs near cluster and split graphs, with tight bounds and kernelization results.

Findings

FPT algorithms for dominating set variants in graphs close to cluster graphs

FPT algorithms for dominating set variants in graphs close to split graphs

Most variants lack polynomial kernels in these graph classes

Abstract

We consider structural parameterizations of the fundamental Dominating Set problem and its variants in the parameter ecology program. We give improved FPT algorithms and lower bounds under well-known conjectures for dominating set in graphs that are k vertices away from a cluster graph or a split graph. These are graphs in which there is a set of k vertices (called the modulator) whose deletion results in a cluster graph or a split graph. We also call k as the deletion distance (to the appropriate class of graphs). When parameterized by the deletion distance k to cluster graphs - we can find a minimum dominating set (DS) in 3^k n^{O(1)}-time. Within the same time, we can also find a minimum independent dominating set (IDS) or a minimum dominating clique (DC) or a minimum efficient dominating set (EDS) or a minimum total dominating set (TDS). We also show that most of these variants of dominating set do not have polynomial sized kernel. Additionally, we show that when parameterized by the deletion distance k to split graphs - IDS can be solved in 2^k n^{O(1)}-time and EDS can be solved in 3^{k/2}n^{O(1)}.