Journal of Applied and Industrial Mathematics

, Volume 11, Issue 1, pp 99–106 | Cite as

Critical elements in combinatorially closed families of graph classes

  • D. S. MalyshevEmail author


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.


computational complexity hereditary class critical class efficient algorithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979; Mir, Moscow, 1982).zbMATHGoogle Scholar
  2. 2.
    D. S. Malyshev, “ContinuumSets of Boundary GraphClasses for the Colorability Problems,” Diskretn. Anal. Issled. Oper. 16 (5), 41–51 (2009).MathSciNetGoogle Scholar
  3. 3.
    D. S. Malyshev, “On Minimal Hard Classes of Graphs,” Diskretn. Anal. Issled. Oper. 16 (6), 43–51 (2009).MathSciNetzbMATHGoogle Scholar
  4. 4.
    D. S. Malyshev, “Classes of Graphs Critical for the Edge List-Ranking Problem,” Diskretn. Anal. Issled. Oper. 20 (6), 59–76 (2013) [J. Appl. Indust. Math. 8 (2), 245–255 (2014)].zbMATHGoogle Scholar
  5. 5.
    V. E. Alekseev, “On Easy and Hard Hereditary Classes of Graphs with Respect to the Independent Set Problem,” Discrete Appl. Math. 132 (1–3), 17–26 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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 (1–2), 219–236 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    V. E. Alekseev, D. V. Korobitsyn, and V. V. Lozin, “Boundary Classes of Graphs for the Dominating Set Problem,” DiscreteMath. 285 (1–3), 1–6 (2004).MathSciNetzbMATHGoogle Scholar
  8. 8.
    S. Arnborg and A. Proskurowski, “Linear Time Algorithms for NP-Hard Problems Restricted to Partial k-Trees,” Discrete Appl. Math. 23 (1), 11–24 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    H. L. Bodlaender, “Dynamic Programming on Graphs with Bounded Treewidth,” in Automata, Languages and Programming (Proceedings of 15th International Colloquium, Tampere, Finland, July 11–15, 1988) (Springer, Heidelberg, 1988), pp. 105–118.CrossRefGoogle Scholar
  10. 10.
    H. L. Bodlaender, “A Partial k-Arboretumof Graphs with Bounded Treewidth,” Theor. Comput. Sci. 209 (1–2), 1–45 (1998).CrossRefzbMATHGoogle Scholar
  11. 11.
    R. Boliac and V. V. Lozin, “On the Clique-Width of Graphs in Hereditary Classes,” in Algorithms and Computation (Proceedings of 13th International Symposium, Vancouver, Canada, November 21–23, 2002) (Springer, Heidelberg, 2002), pp. 44–54.CrossRefGoogle Scholar
  12. 12.
    B. Courcelle, J. Makowsky, and U. Rotics, “Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width,” Theory Comput. Syst. 33 (2), 125–150 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    C. Dubey, U. Feige, and W. Unger, “Hardness Results for Approximating the Bandwidth,” J. Comput. Syst. Sci. 77 (1), 62–90 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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 (Proceedings of 19th International Symposium, Gold Coast, Australia, December 15–17, 2008) (Springer, Heidelberg, 2008), pp. 294–305.CrossRefGoogle Scholar
  15. 15.
    F. Gurski and E. Wanke, “Line Graphs of Bounded Clique-Width,” Discrete Math. 307 (22), 2734–2754 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    D. Kobler and D. Rotics, “Edge Dominating Set and Colorings on Graphs with Fixed Clique-Width,” Discrete Appl. Math. 126 (2–3), 197–221 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Z. Miller, “The Bandwidth of Caterpillar Graphs,” Congr. Numerantium 33, 235–252 (1981).MathSciNetzbMATHGoogle Scholar
  18. 18.
    D. Muradian, “The Bandwidth Minimization Problem for Cyclic Caterpillars with Hair Length 1 is NPComplete,” Theor. Comput. Sci. 307 (3), 567–572 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    N. Robertson and P. Seymour, “Graph Minors V: Excluding a Planar Graph,” J. Combin. Theory Ser. B 41 (1), 92–114 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    N. Robertson and P. Seymour, “GraphMinors XX:Wagner’s Conjecture,” J. Combin. Theory Ser. B 92 (2), 325–357 (2004).MathSciNetCrossRefGoogle Scholar
  21. 21.
    J. B. Saxe, “Dynamic-Programming Algorithms for Recognizing Small-Bandwidth Graphs in Polynomial Time,” SIAM J. Algebraic DiscreteMethods 1, No. 4, 363–369 (1980).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsNizhny NovgorodRussia
  2. 2.Lobachevsky Nizhny Novgorod State UniversityNizhny NovgorodRussia

Personalised recommendations