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A complexity dichotomy and a new boundary class for the dominating set problem

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Abstract

We study the computational complexity of the dominating set problem for hereditary graph classes, i.e., classes of simple unlabeled graphs closed under deletion of vertices. Every hereditary class can be defined by a set of its forbidden induced subgraphs. There are numerous open cases for the complexity of the problem even for hereditary classes with small forbidden structures. We completely determine the complexity of the problem for classes defined by forbidding a five-vertex path and any set of fragments with at most five vertices. Additionally, we also prove polynomial-time solvability of the problem for some two classes of a similar type. The notion of a boundary class is a helpful tool for analyzing the computational complexity of graph problems in the family of hereditary classes. Three boundary classes were known for the dominating set problem prior to this paper. We present a new boundary class for it.

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Acknowledgments

The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2015–2016 (Grant 15-01-0010) and supported within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program.

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  1. National Research University Higher School of Economics, 25/12 Bolshaja Pecherskaja Ulitsa, Nizhny Novgorod, 603155, Russia

    D. S. Malyshev

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  1. D. S. Malyshev
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Malyshev, D.S. A complexity dichotomy and a new boundary class for the dominating set problem. J Comb Optim 32, 226–243 (2016). https://doi.org/10.1007/s10878-015-9872-z

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