Advertisement

Algorithmica

, Volume 71, Issue 2, pp 471–495 | Cite as

Linear-Time Algorithms for Tree Root Problems

  • Maw-Shang Chang
  • Ming-Tat Ko
  • Hsueh-I LuEmail author
Article
  • 257 Downloads

Abstract

Let T be a tree on a set V of nodes. The p-th power T p of T is the graph on V such that any two nodes u and w of V are adjacent in T p if and only if the distance of u and w in T is at most p. Given an n-node m-edge graph G and a positive integer p, the p-th tree root problem asks for a tree T, if any, such that G=T p . Given an n-node m-edge graph G, the tree root problem asks for a positive integer p and a tree T, if any, such that G=T p . Kearney and Corneil gave the best previously known algorithms for both problems. Their algorithm for the former (respectively, latter) problem runs in O(n 3) (respectively, O(n 4)) time. In this paper, we give O(n+m)-time algorithms for both problems.

Keywords

Graph power Graph root Tree power Tree root Chordal graph Maximal clique Minimal node separator 

Notes

Acknowledgements

Maw-Shang Chang thanks the Institute of Information Science of Academia Sinica of Taiwan for their hospitality and support where part of this research took place.

References

  1. 1.
    Agnarsson, G., Halldórsson, M.M.: Coloring powers of planar graphs. SIAM J. Discrete Math. 16(4), 651–662 (2003) zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Agnarsson, G., Halldórsson, M.M.: On colorings of squares of outerplanar graphs. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 244–253 (2004) Google Scholar
  3. 3.
    Agnarsson, G., Halldórsson, M.M.: Strongly simplicial vertices of powers of trees. Discrete Math. 307(21), 2647–2652 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Alon, N., Lubetzky, E.: Graph powers, Delsarte, Hoffman, Ramsey, and Shannon. SIAM J. Discrete Math. 21(2), 329–348 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Alon, N., Lubetzky, E.: Independent sets in tensor graph powers. J. Graph Theory 54(1), 73–87 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Alon, N., Mohar, B.: The chromatic number of graph powers. Comb. Probab. Comput. 11(1), 1–10 (2002) zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Brandstädt, A., Hundt, C., Mancini, F., Wagner, P.: Rooted directed path graphs are leaf powers. Discrete Math. 310(4), 897–910 (2010) zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: a Survey. SIAM, Philadelphia (1999) zbMATHCrossRefGoogle Scholar
  9. 9.
    Buneman, P.: Characterization of rigid circuit graphs. Discrete Math. 9, 205–212 (1974) zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Chandran, L.S., Grandoni, F.: A linear time algorithm to list the minimal separators of chordal graphs. Discrete Math. 306, 351–358 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Chang, M.-S., Ko, M.-T., Lu, H.-I.: Linear-time algorithms for tree root problems. In: Proceedings of the 10th Scandinavian Workshop on Algorithm Theory, pp. 411–422 (2006) Google Scholar
  12. 12.
    Chebikin, D.: Graph powers and k-ordered Hamiltonicity. Discrete Math. 308(15), 3220–3229 (2008) zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, Z.-Z., Jiang, T., Lin, G.: Computing phylogenetic roots with bounded degrees and errors. SIAM J. Comput. 32(4), 864–879 (2003) zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Chiang, Y.-T., Lin, C.-C., Lu, H.-I.: Orderly spanning trees with applications. SIAM J. Comput. 34(4), 924–945 (2005) zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    DeVos, M., McDonald, J., Scheide, D.: Average degree in graph powers. J. Graph Theory 72(1), 7–18 (2013) zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Dinur, I., Friedgut, E., Regev, O.: Independent sets in graph powers are almost contained in juntas. Geom. Funct. Anal. 18(1), 77–97 (2008) zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Dirac, G.A.: On rigid circuit graphs. Abh. Math. Semin. Univ. Hamb. 25, 71–76 (1961) zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Dom, M., Guo, J., Hüffner, F., Niedermeier, R.: Closest 4-leaf power is fixed-parameter tractable. Discrete Appl. Math. 156(18), 3345–3361 (2008) zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Dvořák, Z., Král’, D., Nejedlý, P., Škrekovski, R.: Coloring squares of planar graphs with girth six. Eur. J. Comb. 29(4), 838–849 (2008) zbMATHCrossRefGoogle Scholar
  20. 20.
    Escalante, F., Montejano, L., Rojano, T.: Characterization of n-path graphs and of graphs having nth root. J. Comb. Theory, Ser. B 16, 282–289 (1974) zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Flotow, C.: Graphs whose powers are chordal and graphs whose powers are interval graphs. J. Graph Theory 24(4), 323–330 (1997) zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Gavril, F.: The intersection graphs of subtrees in trees are exactly chordal graphs. J. Comb. Theory, Ser. B 16, 47–56 (1974) zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Geller, D.P.: The square root of a digraph. J. Comb. Theory 5, 320–321 (1968) zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Golumbic, M.: Algorithmic Graph Theory and Perfect Graphs, 2nd edn. North-Holland, Amsterdam (2004) zbMATHGoogle Scholar
  25. 25.
    Gupta, S.K., Singh, A.: On tree roots of graphs. Int. J. Comput. Math. 73, 157–166 (1999) zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Harary, F., Karp, R.M., Tutte, W.T.: A criterion for planarity of the square of a graph. J. Comb. Theory 2, 395–405 (1967) zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    He, X., Kao, M.-Y., Lu, H.-I.: A fast general methodology for information-theoretically optimal encodings of graphs. SIAM J. Comput. 30(3), 838–846 (2000) zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Hell, P., Yu, X., Zhou, H.S.: Independence ratios of graph powers. Discrete Math. 127(1–3), 213–220 (1994) zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Ho, C.-W., Lee, R.C.T.: Counting clique trees and computing perfect elimination schemes in parallel. Inf. Process. Lett. 31, 61–68 (1989) zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Hsu, W.-L., Ma, T.-H.: Fast and simple algorithms for recognizing chordal comparability graphs and interval graphs. SIAM J. Comput. 28, 1004–1020 (1999) zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Jacobson, G.: Space-efficient static trees and graphs. In: Proceedings of the 30th Annual Symposium on Foundations of Computer Science, pp. 549–554 (1989) CrossRefGoogle Scholar
  32. 32.
    Kearney, P.E., Corneil, D.G.: Tree powers. J. Algorithms 29, 111–131 (1998) zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Lau, L.C.: Bipartite roots of graphs. ACM Trans. Algorithms 2(2), 178–208 (2006) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Lau, L.C., Corneil, D.G.: Recognizing powers of proper interval, split, and chordal graphs. SIAM J. Comput. 18(1), 83–102 (2004) zbMATHMathSciNetGoogle Scholar
  35. 35.
    Lin, Y.L., Skiena, S.: Algorithms for square roots of graphs. SIAM J. Discrete Math. 8, 99–118 (1995) zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21, 193–201 (1992) zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Markenzon, L., da Costa Pereira, P.R.: A compact representation for chordal graphs. In: Seventh Cologne Twente Workshop on Graphs and Combinatorial Optimization, pp. 174–176 (2008) Google Scholar
  38. 38.
    Mohar, B., Vodopivec, A.: The genus of Petersen powers. J. Graph Theory 67(1), 1–8 (2011) zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Molloy, M., Salavatipour, M.R.: A bound on the chromatic number of the square of a planar graph. J. Comb. Theory, Ser. B 94(2), 189–213 (2005) zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Motwani, R., Sudan, M.: Computing roots of graphs is hard. Discrete Appl. Math. 54, 81–88 (1994) zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Mukhopadhyay, A.: The square root of a graph. J. Comb. Theory 2, 290–295 (1967) zbMATHCrossRefGoogle Scholar
  42. 42.
    Munro, J.I., Raman, V.: Succinct representation of balanced parentheses, static trees and planar graphs. SIAM J. Comput. 31(3), 762–776 (2001) zbMATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Nishimura, N., Ragde, P., Thilikos, D.M.: On graph powers for leaf-labeled trees. J. Algorithms 42(1), 69–108 (2002) zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Rose, D., Tarjan, R., Lueker, G.: Algorithmic aspects of vertex elimination of graph. SIAM J. Comput. 5(2), 266–283 (1976) zbMATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Ross, I.C., Harary, F.: The squares of a tree. Bell Syst. Tech. J. 39, 641–647 (1960) MathSciNetCrossRefGoogle Scholar
  46. 46.
    Sadakane, K., Navarro, G.: Fully-functional succinct trees. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 134–149 (2010) CrossRefGoogle Scholar
  47. 47.
    Simonyi, G.: Asymptotic values of the Hall-ratio for graph powers. Discrete Math. 306(19–20), 2593–2601 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    Sreenivasa Kumar, P., Veni Madhavan, C.E.: Minimal vertex separators of chordal graphs. Discrete Appl. Math. 89, 155–168 (1998) zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Sreenivasa Kumar, P., Veni Madhavan, C.E.: Clique tree generalization and new subclasses of chordal graphs. Discrete Appl. Math. 117, 109–131 (2002) zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Tarjan, R., Yannakakis, M.: Simple linear time algorithms to test chordality of graphs, test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13(3), 566–576 (1984) zbMATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    van den Heuvel, J., McGuinness, S.: Coloring the square of a planar graph. J. Graph Theory 42(2), 110–124 (2003) zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer Science and Information EngineeringHungkuang UniversityTaichung CityTaiwan
  2. 2.Institute of Information ScienceAcademia SinicaTaipeiTaiwan
  3. 3.Department of Computer Science and Information EngineeringNational Taiwan UniversityTaipeiTaiwan

Personalised recommendations