Abstract
A dominating set S of graph G is called an r-grouped dominating set if S can be partitioned into \(S_1,S_2,\ldots ,S_k\) such that the size of each unit \(S_i\) is r and the subgraph of G induced by \(S_i\) 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 (\(\textsf{MSO}_{2}\)). This fixed-parameter tractability is good news, but the running time is not practical. We then design an \(O^*(\min \{(2\tau (r+1))^{\tau },(2\tau )^{2\tau }\})\)-time algorithm for general \(r\ge 2\), where \(\tau \) 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., \(r \in \{2,3\}\), we can speed up the algorithm, whose running time becomes \(O^*((r+1)^\tau )\). We further argue the relationship between FPT results and graph parameters, which draws the parameterized complexity landscape of r -Grouped Dominating Set.
Partially supported by JSPS KAKENHI Grant Numbers JP17H01698, JP17K19960, JP18H04091, JP20H05793, JP20H05967, JP21K11752, JP21H05852, JP21K17707, JP21K19765, and JP22H00513.
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Notes
- 1.
The \(O^*\) notation suppresses the polynomial factors of the input size.
- 2.
Note that there is no equivalent \(\textsf{MSO}_{1}\) formula of length depending only on r. This is because \(G \models \psi _{2}(V)\) expresses the property of having a perfect matching, for which an \(\textsf{MSO}_{1}\) formula does not exist (see e.g., [10]).
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Hanaka, T., Ono, H., Otachi, Y., Uda, S. (2023). Grouped Domination Parameterized by Vertex Cover, Twin Cover, and Beyond. In: Mavronicolas, M. (eds) Algorithms and Complexity. CIAC 2023. Lecture Notes in Computer Science, vol 13898. Springer, Cham. https://doi.org/10.1007/978-3-031-30448-4_19
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