Parameterized complexity in multiple-interval graphs: domination, partition, separation, irredundancy

TL;DR

This paper investigates the parameterized complexity of various domination, partition, and separation problems in multiple-interval graphs, establishing hardness results and classifications within the W-hierarchy.

Contribution

It provides new complexity classifications for multiple-interval graph problems, answering open questions and contrasting the complexity of related problems.

Findings

k-Dominating Set is W[1]-hard in multiple-interval graphs but belongs to W[1].

d-Distance k-Dominating Set is W[2]-complete for d >= 2.

k-Perfect Code is W[1]-complete even in unit 2-track interval graphs.

Abstract

We show that the problem k-Dominating Set and its several variants including k-Connected Dominating Set, k-Independent Dominating Set, and k-Dominating Clique, when parameterized by the solution size k, are W[1]-hard in either multiple-interval graphs or their complements or both. On the other hand, we show that these problems belong to W[1] when restricted to multiple-interval graphs and their complements. This answers an open question of Fellows et al. In sharp contrast, we show that d-Distance k-Dominating Set for d >= 2 is W[2]-complete in multiple-interval graphs and their complements. We also show that k-Perfect Code and d-Distance k-Perfect Code for d >= 2 are W[1]-complete even in unit 2-track interval graphs. In addition, we present various new results on the parameterized complexities of k-Vertex Clique Partition and k-Separating Vertices in multiple-interval graphs and their complements, and present a very simple alternative proof of the W[1]-hardness of k-Irredundant Set in general graphs.