Kanesh, L., Madathil, J., Roy, S. orcid.org/0000-0003-3633-542X et al. (2 more authors) (2023) Further Exploiting c-Closure for FPT Algorithms and Kernels for Domination Problems. SIAM Journal on Discrete Mathematics, 37 (4). pp. 2626-2669. ISSN 0895-4801
Abstract
For a positive integer c, a graph G is said to be c-closed if every pair of nonadjacent vertices in G have at most c - 1 neighbors in common. The closure of a graph G, denoted by cl(G), is the least positive integer c for which G is c-closed. The class of c-closed graphs was introduced by J. Fox, T. Roughgarden, C. Seshadhri, F. Wei, and N. Wein [Proceedings of the International Colloquium on Automata, Languages, and Programming (2018), 55; SIAM J. Comput., 49 (2020), pp. 448-464]. T. Koana, C. Komusiewicz, and F. Sommer [Proceedings of the European Symposium on Algorithms (2020), 65; SIAM J. Discrete Math., 36 (2022), pp. 2798-2821] started the study of using cl(G) as an additional structural parameter to design kernels for problems that are W-hard under standard parameterizations. In particular, they studied problems such as Independent Set, Induced Matching, Irredundant Set, and (Threshold) Dominating Set and showed that each of these problems admits a polynomial kernel when parameterized either by k + c or by k for each fixed value of c. Here, k is the solution size and c = cl(G). The work of Koana et al. left several questions open, one of which was whether the Perfect Code problem admits a fixed-parameter tractable (FPT) algorithm and a polynomial kernel on c-closed graphs. In this paper, among other results, we answer this question in the affirmative. Inspired by the FPT algorithm for Perfect Code, we further explore two more domination problems on the graphs of bounded closure. The other problems that we study are Connected Dominating Set and Partial Dominating Set. We show that Perfect Code and Connected Dominating Set are fixed-parameter tractable when parameterized by k + cl(G), whereas Partial Dominating Set, parameterized by k, is W[1]-hard even when cl(G) = 2. We also show that for each fixed c, Perfect Code admits a polynomial kernel on the class of c-closed graphs. And we observe that Connected Dominating Set has no polynomial kernel even on 2-closed graphs unless NP ⊆ co-NP/poly.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2023 Lawqueen Kanesh, Jayakrishnan Madathil, Sanjukta Roy, Abhishek Sahu, and Saket Saurabh. This is an author produced version of an article published in SIAM Journal on Discrete Mathematics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | c-closed graphs; domination problems; perfect code; connected dominating set; fixed-parameter tractable; polynomial kernel |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 14 Feb 2024 16:19 |
Last Modified: | 14 Feb 2024 16:19 |
Status: | Published |
Publisher: | Society for Industrial & Applied Mathematics |
Identification Number: | 10.1137/22m1491721 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:209176 |