Complexity of Paired Domination in AT-free and Planar Graphs

Tripathi, Vikash; Kloks, Ton; Pandey, Arti; Paul, Kaustav; Wang, Hung-Lung

doi:10.1007/978-3-030-95018-7_6

Cited by 4 publications

(3 citation statements)

References 13 publications

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“…We show that r-Grouped Dominating Set is NP-hard even on planar bipartite graphs of maximum degree 3 for every fixed r ≥ 1. This strengthens the result by Tripathi et al [36].…”

Section: Basic Resultssupporting

confidence: 91%

“…Tripathi et al [36] showed that Paired Dominating Set (equivalently, 2-Grouped Dominating Set) is NP-complete for planar graphs with maximum degree 5. We show that r-Grouped Dominating Set is NP-hard even on planar bipartite graphs of maximum degree 3 for every fixed r ≥ 1.…”

Section: Basic Resultsmentioning

confidence: 99%

“…The notion of the paired dominating set is introduced in [25,26] by Haynes and Slater as a model of dominating sets with pairwise backup. It is NP-hard on split graphs, bipartite graphs [10], graphs of maximum degree 3 [8], and planar graphs of maximum degree 5 [36], whereas it can be solved in polynomial time on strongly-chordal graphs [9], distance-hereditary graphs [30], and AT-free graphs [36]. There are several graph classes (e.g., strongly orderable graphs [33]) where the paired dominating set problem is tractable, whereas the ordinary dominating set problem remains NP-hard.…”

Section: Related Workmentioning

confidence: 99%

See 2 more Smart Citations

Grouped Domination Parameterized by Vertex Cover, Twin Cover, and Beyond

Hanaka¹,

Ono²,

Otachi³

et al. 2023

Preprint

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A dominating set S of graph G is called an r-grouped dominating set if S can be partitioned into S1, S2, . . . , S k such that the size of each unit Si is r and the subgraph of G induced by Si is connected. The concept of r-grouped dominating sets generalizes several well-studied variants of dominating sets with requirements for connected component sizes, such as the ordinary dominating sets (r = 1), paired dominating sets (r = 2), and connected dominating sets (r is arbitrary and k = 1). In this paper, we investigate the computational complexity of r-Grouped Dominating Set, which is the problem of deciding whether a given graph has an r-grouped dominating set with at most k units. For general r, r-Grouped Dominating Set is hard to solve in various senses because the hardness of the connected dominating set is inherited. We thus focus on the case in which r is a constant or a parameter, but we see that r-Grouped Dominating Set for every fixed r > 0 is still hard to solve. From the observations about the hardness, we consider the parameterized complexity concerning well-studied graph structural parameters. We first see that r-Grouped Dominating Set is fixed-parameter tractable for r and treewidth, which is derived from the fact that the condition of r-grouped domination for a constant r can be represented as monadic second-order logic (MSO2). This fixed-parameter tractability is good news, but the running time is not practical. We then design an O * (min{(2τ (r + 1)) τ , (2τ ) 2τ })-time algorithm for general r ≥ 2, where τ is the twin cover number, which is a parameter between vertex cover number and clique-width. For paired dominating set and trio dominating set, i.e., r ∈ {2, 3}, we can speed up the algorithm, whose running time becomes O * ((r + 1) τ ). We further argue the relationship between FPT results and graph parameters, which draws the parameterized complexity landscape of r-Grouped Dominating Set.

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“…We show that r-Grouped Dominating Set is NP-hard even on planar bipartite graphs of maximum degree 3 for every fixed r ≥ 1. This strengthens the result by Tripathi et al [36].…”

Section: Basic Resultssupporting

confidence: 91%

Section: Basic Resultsmentioning

confidence: 99%