Theoretical Computer Science ( IF 0.747 ) Pub Date : 2020-02-11 , DOI: 10.1016/j.tcs.2020.02.014
Mingyu Xiao, Zimo Sheng

A mixed domination of a graph $G=\left(V,E\right)$ is a mixed set D of vertices and edges such that for every edge or vertex, if it is not in D, then it is adjacent or incident to at least one vertex or edge in D. The Mixed Domination problem is to check whether there is a mixed domination of size at most k in a graph. Mixed domination is a mixture concept of vertex domination and edge domination, and the mixed domination problem has been studied from the view of approximation algorithms, parameterized algorithms, and so on. In this paper, we give a branch-and-search algorithm with running time bound of ${O}^{⁎}\left({4.172}^{k}\right)$, which improves the previous bound of ${O}^{⁎}\left({7.465}^{k}\right)$. For kernelization, it is known that the problem parameterized by k in general graphs is unlikely to have a polynomial kernel. We show the problem in planar graphs allows linear kernel by giving a kernel of $11k-16$ vertices.

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