On the Parameterized Complexity of Reconfiguration of Connected Dominating Sets
Abstract
In a reconfiguration version of an optimization 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 stepbystep transformation between $S$ and $T$ such that all intermediate steps also constitute feasible solutions. In this work, we study the parameterized complexity of the \textsc{Connected Dominating Set Reconfiguration} problem (\textsc{CDSR)}. It was shown in previous work that the \textsc{Dominating Set Reconfiguration} problem (\textsc{DSR}) parameterized by $k$, the maximum allowed size of a dominating set in a reconfiguration sequence, is fixedparameter tractable on all graphs that exclude a biclique $K_{d,d}$ as a subgraph, for some constant $d \geq 1$. We show that the additional connectivity constraint makes the problem much harder, namely, that \textsc{CDSR} is \textsf{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 \textsc{CDSR} parameterized by $k$ is fixedparameter tractable, and in fact admits a polynomial kernel on planar graphs.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.00581
 Bibcode:
 2019arXiv191000581L
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics;
 Mathematics  Combinatorics