English version:
Journal of Applied and Industrial Mathematics, 2017, 11:1, 99-106

Volume 24, No 1, 2017, P. 81-96

UDC 519.17
D. S. Malyshev
Critical elements in combinatorially closed families of graph classes

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.
Bibliogr. 21.

Keywords: computational complexity, hereditary class, critical class, efficient algorithm.

DOI: 10.17377/daio.2017.24.523

Dmitry S. Malyshev 1,2
1. National Research University Higher School of Economics,
25/12 Bolshaya Pecherskaya St., 603155 Nizhny Novgorod, Russia
2. Lobachevsky State University,
23 Gagarin Ave., 603950 Nizhny Novgorod, Russia
e-mail: dmalishev@hse.ru, dsmalyshev@rambler.ru

Received 11 January 2016
Revised 29 April 2016


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