On the Parameterized Complexity of Reconfiguration of Connected Dominating Sets

Daniel Lokshtanov,Fahad Panolan,Amer E Mouawad,Sebastian Siebertz

doi:10.1007/s00453-021-00909-5

Abstract

In a reconfiguration version of a decision problem \(\mathcal {Q}\) the input is an instance of \(\mathcal {Q}\) and two feasible solutions S and T. The objective is to determine whether there exists a step-by-step transformation between S and T such that all intermediate steps also constitute feasible solutions. In this work, we study the parameterized complexity of the Connected Dominating Set Reconfiguration problem (CDS-R). It was shown in previous work that the Dominating Set Reconfiguration problem (DS-R) parameterized by k, the maximum allowed size of a dominating set in a reconfiguration sequence, is fixed-parameter tractable on all graphs that exclude a biclique \(K_{d,d}\) as a subgraph, for some constant \(d \ge 1\). We show that the additional connectivity constraint makes the problem much harder, namely, that CDS-R is W[1]-hard parameterized by \(k+\ell \), the maximum allowed size of a dominating set plus the length of the reconfiguration sequence, already on 5-degenerate graphs. On the positive side, we show that CDS-R parameterized by k is fixed-parameter tractable, and in fact admits a polynomial kernel on planar graphs.

Highlights

In a decision problem Q, we are usually asked to determine the existence of a feasible solution for an instance I of Q
We consider a graph, called the reconfiguration graph, that has one vertex for each feasible solution and where two vertices are connected by an edge if we allow the transformation between the two Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 24:2 Reconfiguration of Connected Dominating Set corresponding solutions
The existence of a reconfiguration sequence of length at most with connected dominating sets of size at most k can be expressed by a first-order formula of length depending only on k and

Summary

Introduction

In a decision problem Q, we are usually asked to determine the existence of a feasible solution for an instance I of Q. Both pathwidth and treewidth of a graph are bounded by its bandwidth, the Dominating Set Reconfiguration problem is PSPACE-complete on graphs of bounded pathwidth and treewidth These hardness results motivated the study of the parameterized complexity of the problem. As 5-degenerate graphs exclude the biclique K6,6 as a subgraph, Dominating Set Reconfiguration is fixed-parameter tractable on much more general graph classes than its connected variant. The existence of a reconfiguration sequence of length at most with connected dominating sets of size at most k can be expressed by a first-order formula of length depending only on k and It follows from [13] that the problem is fixed-parameter tractable parameterized by k + on every nowhere dense graph class and the same is implied by [2] for every class of bounded cliquewidth. Due to space constraints proofs of results marked with a are deferred to the full version of the paper

Preliminaries

Hardness on degenerate graphs

Fixed-parameter tractability on planar graphs

Domination cores

Reduction rules