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Critical elements in combinatorially closed families of graph classes

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Abstract

The notions of boundary and minimal hard classes of graphs, united by the term “critical classes,” are useful tools for analysis of computational complexity of graph problems in the family of hereditary graph classes. In this family, boundary classes are known for several graph problems. In the paper, we consider critical graph classes in the families of strongly hereditary and minor closed graph classes. Prior to our study, there was the only one example of a graph problem for which boundary classes were completely described in the family of strongly hereditary classes. Moreover, no boundary classes were known for any graph problem in the family of minor closed classes. In this article, we present several complete descriptions of boundary classes for these two families and some classical graph problems. For the problem of 2-additive approximation of graph bandwidth, we find a boundary class in the family of minor closed classes. Critical classes are not known for this problem in the other two families of graph classes.

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Authors and Affiliations

  1. National Research University Higher School of Economics, ul. Bol’shaya Pecherskaya 25/12, Nizhny Novgorod, 603155, Russia

    D. S. Malyshev

  2. Lobachevsky Nizhny Novgorod State University, pr. Gagarina 23, Nizhny Novgorod, 603950, Russia

    D. S. Malyshev

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  1. D. S. Malyshev
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Correspondence to D. S. Malyshev.

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Original Russian Text © D.S. Malyshev, 2017, published in Diskretnyi Analiz i Issledovanie Operatsii, 2017, Vol. 24, No. 1, pp. 81–96.

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Malyshev, D.S. Critical elements in combinatorially closed families of graph classes. J. Appl. Ind. Math. 11, 99–106 (2017). https://doi.org/10.1134/S1990478917010112

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  • DOI: https://doi.org/10.1134/S1990478917010112

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