Graphs and Combinatorics

, 25:557 | Cite as

On the Structure of Contractible Edges in k-connected Partial k-trees

  • N. S. Narayanaswamy
  • N. Sadagopan
  • L. Sunil Chandran
Article

Abstract

Contraction of an edge e merges its end points into a new single vertex, and each neighbor of one of the end points of e is a neighbor of the new vertex. An edge in a k-connected graph is contractible if its contraction does not result in a graph with lesser connectivity; otherwise the edge is called non-contractible. In this paper, we present results on the structure of contractible edges in k-trees and k-connected partial k-trees. Firstly, we show that an edge e in a k-tree is contractible if and only if e belongs to exactly one (k + 1) clique. We use this characterization to show that the graph formed by contractible edges is a 2-connected graph. We also show that there are at least |V(G)| + k − 2 contractible edges in a k-tree. Secondly, we show that if an edge e in a partial k-tree is contractible then e is contractible in any k-tree which contains the partial k-tree as an edge subgraph. We also construct a class of contraction critical 2k-connected partial 2k-trees.

Keywords

Connectivity Contraction Contractible edges Partial k-trees 

References

  1. Ando, K., Kaneko, K., Kawarabayashi, A.: Vertices of degree 5 in a 5-contraction critical graph. Graphs Combin 21(1), 27–37 (2005)Google Scholar
  2. Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math. 23, 11–24 (1989)Google Scholar
  3. Dean, N., Hemminger, R.L., Toft, B.: On contractible edges in 3-connected graphs. Congr. Numer. 58, 291–293 (1987)Google Scholar
  4. Dean, N., Hemminger, R.L., Ota, K.: k Longest cycles in 3-connected graphs contain three contractible edges. J. Graph Theory 13(1), 17–21 (1989)Google Scholar
  5. Dean, N.: Distribution of contractible edges in k-connected graphs. J. Combin. Theory, Ser. B, 48, 1–5 (2003)Google Scholar
  6. Ellingham, M.N., Hemminger, R.L., Johnson, K.E.: Contractible edges in longest cycles in non-hamiltonian graphs. Discrete Math. 133 89–98 (1994)Google Scholar
  7. Fujita, K.: Maximum number of contractible edges on Hamiltonian cycles of a 3-connected graph. Graphs Combin. 18(3), 447–478 (2005)Google Scholar
  8. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic, London (1980)Google Scholar
  9. Jani, M., Rieger, R.G., Zeleke, M.: Enumeration of k-trees and applications. Ann. Combin. 6, 375–382 (2002)Google Scholar
  10. Kriesell, M.: A survey on contractible edges in graphs of a prescribed vertex connectivity. Graphs Combin. 18, 1–30 (2002)Google Scholar
  11. Mandaltsis, D., Kontoleon, J.M.: Enumeration of k-trees and their applications to the reliability evaluation of communication networks. Ann. Combin. 29, 733–735 (1989)Google Scholar
  12. Martinov, N.: Uncontractible 4-connected graphs. J. Graph Theory 6, 343–344 (1982)Google Scholar
  13. Martinov, N.: A recursive characterisation of 4-connected graphs. Discrete Math. 84, 105–108 (1990)Google Scholar
  14. Proskurowski, A.: Recursive graphs, recursive labelings and shortest paths. SIAM J. Comput. 10(2), 391–397 (1981)Google Scholar
  15. Saito, A.: Covering contractible edges in 3-connected graphs. J. Graph Theory 14(6), 635–643 (1990)Google Scholar
  16. Saito, A., Ando, K., Enomoto, H.: Contractible edges in 3-connected graphs. J. Combin. Theory-B 42(1) (1987)Google Scholar
  17. Rose, D.J.: Triangulated graphs and elimination process. J. Math. Anal. Appl. 32, 597–609 (1970)Google Scholar
  18. Rose, D.J.: On simple characterisations of k-trees. Discrete Math. 7, 317–322 (1974)Google Scholar
  19. Thomassen, C.: Non-separating cycles in k-connected graphs. J. Graph Theory 5, 351–354 (1981)Google Scholar
  20. Tutte, W.T.: A theory of 3-connected graphs. Indag. Math 23, 441–455 (1961)Google Scholar
  21. West, D.B.: Introduction to graph theory. Prentice Hall of India (2003)Google Scholar
  22. Yang, D., Iwama, K., Naoki, K.: A new probabilistic analysis of Karger’s randomized algorithm for min-cut problems. Inf. Process. Lett. 64(5), 255–261 (1997)Google Scholar

Copyright information

© Springer 2009

Authors and Affiliations

  • N. S. Narayanaswamy
    • 1
  • N. Sadagopan
    • 1
  • L. Sunil Chandran
    • 2
  1. 1.Department of Computer Science and EngineeringIndian Institute of TechnologyChennaiIndia
  2. 2.Computer Science and AutomationIndian Institute of ScienceBangaloreIndia

Personalised recommendations