Star Partitions of Perfect Graphs

  • René van Bevern
  • Robert Bredereck
  • Laurent Bulteau
  • Jiehua Chen
  • Vincent Froese
  • Rolf Niedermeier
  • Gerhard J. Woeginger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)


The partition of graphs into nice subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into stars, a problem known to be NP-complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP-hard cases, for example, on grid graphs and chordal graphs.


Bipartite Graph Planar Graph Token List Interval Graph Chordal Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asdre, K., Nikolopoulos, S.D.: NP-completeness results for some problems on subclasses of bipartite and chordal graphs. Theor. Comput. Sci. 381(1-3), 248–259 (2007)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Berman, F., Johnson, D., Leighton, T., Shor, P.W., Snyder, L.: Generalized planar matching. J. Algorithms 11(2), 153–184 (1990)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    van Bevern, R., Bredereck, R., Chen, J., Froese, V., Niedermeier, R., Woeginger, G.J.: Star partitions of perfect graphs, TU Berlin (2014a) (manuscript) arXiv:1402.2589 [cs.DM]Google Scholar
  4. 4.
    van Bevern, R., Bredereck, R., Chen, J., Froese, V., Niedermeier, R., Woeginger, G.J.: Network-based dissolution, TU Berlin (2014b) (manuscript) arXiv:1402.2664 [cs.DM] Google Scholar
  5. 5.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: a Survey. In: SIAM Monographs on Discrete Mathematics and Applications, vol. 3. SIAM (1999)Google Scholar
  6. 6.
    Chalopin, J., Paulusma, D.: Packing bipartite graphs with covers of complete bipartite graphs. Discrete Appl. Math. 168, 40–50 (2014)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Corneil, D., Perl, Y., Stewart, L.: A linear recognition algorithm for cographs. SIAM J. Comput. 14(4), 926–934 (1985)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Cornuéjols, G.: General factors of graphs. J. Combin. Theory Ser. B 45(2), 185–198 (1988)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Dahlhaus, E., Karpinski, M.: Matching and multidimensional matching in chordal and strongly chordal graphs. Discrete Appl. Math. 84(1-3), 79–91 (1998)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    De Bontridder, K.M.J., Halldórsson, B.V., Halldórsson, M.M., Hurkens, C.A.J., Lenstra, J.K., Ravi, R., Stougie, L.: Approximation algorithms for the test cover problem. Math. Program. 98(1-3), 477–491 (2003)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Dyer, M.E., Frieze, A.M.: On the complexity of partitioning graphs into connected subgraphs. Discrete Appl. Math. 10(2), 139–153 (1985)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics. Elsevier, Amsterdam (2004)MATHGoogle Scholar
  13. 13.
    Kirkpatrick, D.G., Hell, P.: On the complexity of general graph factor problems. SIAM J. Comput. 12(3), 601–608 (1983)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Kosowski, A., Małafiejski, M., Żyliński, P.: Parallel processing subsystems with redundancy in a distributed environment. In: Wyrzykowski, R., Dongarra, J., Meyer, N., Waśniewski, J. (eds.) PPAM 2005. LNCS, vol. 3911, pp. 1002–1009. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Małafiejski, M., Żyliński, P.: Weakly cooperative guards in grids. In: Gervasi, O., Gavrilova, M.L., Kumar, V., Laganá, A., Lee, H.P., Mun, Y., Taniar, D., Tan, C.J.K. (eds.) ICCSA 2005. LNCS, vol. 3480, pp. 647–656. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Monnot, J., Toulouse, S.: The path partition problem and related problems in bipartite graphs. Oper. Res. Lett. 35(5), 677–684 (2007)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    van Rooij, J.M.M., van Kooten Niekerk, M.E., Bodlaender, H.L.: Partition into triangles on bounded degree graphs. Theory Comput. Syst. 52(4), 687–718 (2013)MATHMathSciNetGoogle Scholar
  18. 18.
    Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Comput. Geom. 1(1), 343–353 (1986)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Spinrad, J., Brandstädt, A., Stewart, L.: Bipartite permutation graphs. Discrete Appl. Math. 18(3), 279–292 (1987)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Steiner, G.: On the k-path partition problem in cographs. Congressus Numerantium 147, 89–96 (2000)MATHMathSciNetGoogle Scholar
  21. 21.
    Steiner, G.: On the k-path partition of graphs. Theor. Comput. Sci. 290(3), 2147–2155 (2003)CrossRefMATHGoogle Scholar
  22. 22.
    Takamizawa, K., Nishizeki, T., Saito, N.: Linear-time computability of combinatorial problems on series-parallel graphs. J. ACM 29(3), 623–641 (1982)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Yan, J.-H., Chen, J.-J., Chang, G.J.: Quasi-threshold graphs. Discrete Appl. Math. 69(3), 247–255 (1996)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Yan, J.-H., Chang, G.J., Hedetniemi, S.M., Hedetniemi, S.T.: k-path partitions in trees. Discrete Appl. Math. 78(1-3), 227–233 (1997)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Yuster, R.: Combinatorial and computational aspects of graph packing and graph decomposition. Computer Science Review 1(1), 12–26 (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • René van Bevern
    • 1
  • Robert Bredereck
    • 1
  • Laurent Bulteau
    • 1
  • Jiehua Chen
    • 1
  • Vincent Froese
    • 1
  • Rolf Niedermeier
    • 1
  • Gerhard J. Woeginger
    • 2
  1. 1.Institut für Softwaretechnik und Theoretische InformatikTU BerlinGermany
  2. 2.Department of Mathematics and Computer ScienceTU EindhovenThe Netherlands

Personalised recommendations