Abstract
A mixed dominating set (mds) of a graph G is a set \(S \subseteq V(G) \cup E(G)\) such that every element \(x \in (V(G) \cup E(G)) \setminus S\) is either adjacent to or incident with an element of S. In the Mixed Dominating Set (MDS) problem, we are given an n-vertex graph G and a positive integer k, and the objective is to decide whether G has an mds of size at most k. On general graphs, MDS parameterized by k is fixed-parameter tractable, but has no polynomial kernel unless coNP \(\subseteq \) NP/Poly. In this paper, we study the restriction of MDS to several graph classes and establish the following results.
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On proper interval graphs, MDS is polynomial time solvable.
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On graphs that exclude \(K_{d,d}\) as a subgraph, MDS admits a kernel of size \(\mathcal {O}(k^d)\).
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On split graphs, MDS does not admit a polynomial kernel unless coNP \(\subseteq \) NP/Poly.
In addition, we show that on general graphs, MDS admits an exact algorithm with running time \(2^n n^{\mathcal {O}(1)}\).
This work is supported by the European Research Council (ERC) via grant LOPPRE, reference 819416 and Norwegian Research Council via project MULTIVAL.
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- 1.
\(\mathcal {O}^\star \) notation suppresses the polynomial factor. That is, \(\mathcal {O}(f(k)n^{\mathcal {O}(1)})=\mathcal {O}^\star (f(k))\).
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Madathil, J., Panolan, F., Sahu, A., Saurabh, S. (2019). On the Complexity of Mixed Dominating Set. In: van Bevern, R., Kucherov, G. (eds) Computer Science – Theory and Applications. CSR 2019. Lecture Notes in Computer Science(), vol 11532. Springer, Cham. https://doi.org/10.1007/978-3-030-19955-5_23
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