Advertisement

Polynomial Time Recognition of Squares of Ptolemaic Graphs and 3-sun-free Split Graphs

  • Van Bang LeEmail author
  • Andrea Oversberg
  • Oliver Schaudt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

Abstract

The square of a graph \(G\), denoted \(G^2\), is obtained from \(G\) by putting an edge between two distinct vertices whenever their distance is two. Then \(G\) is called a square root of \(G^2\). Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a chordal graph or even a split graph.

We present a polynomial time algorithm that decides whether a given graph has a ptolemaic square root. If such a root exists, our algorithm computes one with a minimum number of edges.

In the second part of our paper, we give a characterization of the graphs that admit a 3-sun-free split square root. This characterization yields a polynomial time algorithm to decide whether a given graph has such a root, and if so, to compute one.

Keywords

Square of graph Square of ptolemaic graph Square of split graph Recognition algorithm 

References

  1. 1.
    Adamszek, A., Adamszek, M.: Large-girth roots of graphs. SIAM J. Discrete Math. 24, 1501–1514 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adamszek, A., Adamszek, M.: Uniqueness of graph square roots of girth six. Electron. J. Combin. 18, 139 (2011)MathSciNetGoogle Scholar
  3. 3.
    Bandelt, H.-J., Henkmann, A., Nicolai, F.: Powers of distance-hereditary graphs. Disc. Math. 145, 37–60 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bandelt, H.-J., Mulder, H.M.: Distance-hereditary graphs. J. Combin. Theory (Ser. B) 41, 182–208 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chang, M.-S., Ko, M.-T., Lu, H.-I.: Linear-time algorithms for tree root problems. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 411–422. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Cochefert, M., Couturier, J.-F., Golovach, P.A., Kratsch, D., Paulusma, D.: Sparse square roots. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds.) WG 2013. LNCS, vol. 8165, pp. 177–188. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Dalhaus, E., Duchet, P.: On strongly chordal graphs. Ars Combin. 24B, 23–30 (1987)Google Scholar
  9. 9.
    Dourado, M.C., Protti, F., Szwarcfiter, J.L.: Complexity aspects of the helly property: graphs and hypergraphs. Electron. J. Combin. 17, 1–53 (2009)CrossRefGoogle Scholar
  10. 10.
    Farzad, B., Karimi, M.: Square-root finding problem in graphs, a complete dichotomy theorem, arXiv:1210.7684 (2012)
  11. 11.
    Farzad, B., Lau, L.C., Le, V.B., Tuy, N.N.: Complexity of finding graph roots with girth conditions. Algorithmica 62, 38–53 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hammer, P.L., Maffray, F.: Completely separable graphs. Disc. App. Math. 27, 85–99 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Howorka, E.: A characterization of ptolemaic graphs. J. Graph Theory 5, 323–331 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lau, L.C.: Bipartite roots of graphs. ACM Trans. Algorithms 2, 178–208 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lau, L.C., Corneil, D.G.: Recognizing powers of proper interval, split, and chordal graphs. SIAM J. Discrete Math. 18, 83–102 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Le, V.B., Nguyen, N.T.: Hardness results and efficient algorithms for graph powers. In: Paul, C., Habib, M. (eds.) WG 2009. LNCS, vol. 5911, pp. 238–249. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Le, V.B., Tuy, N.N.: The square of a block graph. Disc. Math. 310, 734–741 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Le, V.B., Tuy, N.N.: A good characterization of squares of strongly chordal split graphs. Inf. Process. Lett. 310, 120–123 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lubiw, A.: Doubly lexical orderings of matrices. SIAM J. Comput. 16, 854–879 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Milanič, M., Schaudt, O.: Computing square roots of trivially perfect and threshold graphs. Disc. App. Math. 161, 1538–1545 (2013)CrossRefzbMATHGoogle Scholar
  21. 21.
    Motwani, R., Sudan, M.: Computing roots of graphs is hard. Disc. App. Math. 54, 81–88 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Prisner, E.: Hereditary clique-Helly graphs. J. Comb. Math. Comb. Comput. 14, 216–220 (1993)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Raychaudhuri, A.: On powers of strongly chordal and circular arc graphs. Ars Combin. 34, 147–160 (1992)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM J. Comput. 6, 505–517 (1977)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Van Bang Le
    • 1
    Email author
  • Andrea Oversberg
    • 2
  • Oliver Schaudt
    • 2
  1. 1.Institut Für InformatikUniversität RostockRostockGermany
  2. 2.Institut Für InformatikUniversität zu KölnKölnGermany

Personalised recommendations