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On the Parameterized Complexity of Reconfiguration of Connected Dominating Sets

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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.

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Notes

  1. We note that the problem is easily shown to be slicewise polynomial parameterized for parameter \(k + \ell \) as one can guess each set in the reconfiguration sequence.

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Authors and Affiliations

  1. University of California Santa Barbara, Santa Barbara, USA

    Daniel Lokshtanov

  2. Department of Computer Science, American University of Beirut, Beirut, Lebanon

    Amer E. Mouawad

  3. Department of Computer Science and Engineering, IIT Hyderabad, Hyderabad, India

    Fahad Panolan

  4. University of Bremen, Bremen, Germany

    Sebastian Siebertz

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  1. Daniel Lokshtanov
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  2. Amer E. Mouawad
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  3. Fahad Panolan
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  4. Sebastian Siebertz
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Correspondence to Amer E. Mouawad.

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A preliminary version of this paper was accepted for publication at the 15th International Symposium on Parameterized and Exact Computation, IPEC 2020, December 14–18, 2020, Hong Kong, China. The second author is supported by URB project “A theory of change through the lens of reconfiguration”.

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Lokshtanov, D., Mouawad, A.E., Panolan, F. et al. On the Parameterized Complexity of Reconfiguration of Connected Dominating Sets. Algorithmica 84, 482–509 (2022). https://doi.org/10.1007/s00453-021-00909-5

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  • DOI: https://doi.org/10.1007/s00453-021-00909-5

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