EN|RU
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

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.
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

Revised 29 April 2016

# References

[1] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979. Translated under the title Vychislitel’nye mashiny i trudnoreshaemye zadachi, Mir, Moscow, 1982 [Russian].

[2] D. S. Malyshev, Continuum sets of boundary graph classes for the colorability problems, Diskretn. Anal. Issled. Oper., 16, No. 5, 41–51, 2009 [Russian].

[3] D. S. Malyshev, On minimal hard classes of graphs, Diskretn. Anal. Issled. Oper., 16, No. 6, 43–51, 2009 [Russian].

[4] D. S. Malyshev, Classes of graphs critical for the edge list-ranking problem, Diskretn. Anal. Issled. Oper., 20, No. 6, 59–76, 2013 [Russian]. Translated in J. Appl. Ind. Math., 8, No. 2, 245–255, 2014.

[5] V. E. Alekseev, On easy and hard hereditary classes of graphs with respect to the independent set problem, Discrete Appl. Math., 132, No. 1–3, 17–26, 2003.

[6] V. E. Alekseev, R. Boliac, D. V. Korobitsyn, and V. V. Lozin,NP-hard graph problems and boundary classes of graphs, Theor. Comput. Sci., 389, No. 1–2, 219–236, 2007.

[7] V. E. Alekseev, D. V. Korobitsyn, and V. V. Lozin, Boundary classes of graphs for the dominating set problem, Discrete Math., 285, No. 1–3, 1–6, 2004.

[8] S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial k-trees, Discrete Appl. Math., 23, No. 1, 11–24, 1989.

[9] H. L. Bodlaender, Dynamic programming on graphs with bounded treewidth, in Automata, Languages and Programming (Proc. 15th Int. Colloq., Tampere, Finland, July 11–15, 1988), pp. 105–118, Springer-Verlag, Heidelberg, 1988 (Lect. Notes Comput. Sci., Vol. 317).

[10] H. L. Bodlaender, A partial $k$-arboretum of graphs with bounded treewidth, Theor. Comput. Sci., 209, No. 1–2, 1–45, 1998.

[11] R. Boliac and V. V. Lozin, On the clique-width of graphs in hereditary classes, in Algorithms and Computation (Proc. 13th Int. Symp., Vancouver, Canada, Nov. 21–23, 2002), pp. 44–54, Springer, Heidelberg, 2002 (Lect. Notes Comput. Sci., Vol. 2518).

[12] B. Courcelle, J. Makowsky, and U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, Theory Comput. Syst., 33, No. 2, 125–150, 2000.

[13] C. Dubey, U. Feige, and W. Unger, Hardness results for approximating the bandwidth, J. Comput. Syst. Sci., 77, No. 1, 62–90, 2011.

[14] M. R. Fellows, D. Lokshtanov, N. Misra, F. A. Rosamond, and S. Saurabh, Graph layout problems parameterized by vertex cover, in Algorithms and Computation (Proc. 19th Int. Symp., Gold Coast, Australia, Dec. 15–17, 2008), pp. 294–305, Springer, Heidelberg, 2008 (Lect. Notes Comput. Sci., Vol. 5369).

[15] F. Gurski and E. Wanke, Line graphs of bounded clique-width, Discrete Math., 307, No. 22, 2734–2754, 2007.

[16] D. Kobler and D. Rotics, Edge dominating set and colorings on graphs with fixed clique-width, Discrete Appl. Math., 126, No. 2–3, 197–221, 2003.

[17] Z. Miller, The bandwidth of caterpillar graphs, Congr. Numerantium, 33, 235–252, 1981.

[18] D. Muradian, The bandwidth minimization problem for cyclic caterpillars with hair length 1 is NP-complete, Theor. Comput. Sci., 307, No. 3, 567–572, 2003.

[19] N. Robertson and P. Seymour, Graph minors V: Excluding a planar graph, J. Comb. Theory, Ser. B, 41, No. 1, 92–114, 1986.

[20] N. Robertson and P. Seymour, Graph minors XX: Wagner’s conjecture, J. Comb. Theory, Ser. B, 92, No. 2, 325–357, 2004.

[21] J. B. Saxe, Dynamic-programming algorithms for recognizing small-bandwidth graphs in polynomial time, SIAM J. Algebraic Discrete Methods, 1, No. 4, 363–369, 1980.
© Sobolev Institute of Mathematics, 2015