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A Cycle Detection-Based Efficient Approach for Constructing Spanning Trees Directly from Nonregular Graphic Sequences

  • Prantik BiswasEmail author
  • Shahin Shabnam
  • Abhisek Paul
  • Paritosh Bhattacharya
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 394)

Abstract

Realization of graphic sequences and finding the spanning tree of a graph are two popular problems of combinatorial optimization. A simple graph that realizes a given nonnegative integer sequence is often termed as a realization of the given sequence. In this paper, we have proposed a method for obtaining a spanning tree directly from a degree sequence by applying cycle detection algorithm, provided the degree sequence is graphic and nonregular. The proposed method is a two-step process. First, we apply an algorithm to check whether the input sequence is realizable through the construction of an adjacency matrix corresponding to the degree sequence. Then we apply the cycle detection algorithm separately to generate the spanning tree from it.

Keywords

Spanning tree Graph Algorithms Cycle detection Graphic realization Degree sequence Adjacency matrix 

References

  1. 1.
    Kruskal JB. On the shortest spanning subtree of a graph and the travelling salesman problem. Proc Am Math Soc. 1956;7:48–50.CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Prim RC. Shortest connection networks and some generalizations. Bell Syst Tech J. 1957;4:53–7.Google Scholar
  3. 3.
    Havel V. A remark on the existence of finite graphs. (Czech.) Casopis Pest Mat. 1955;80:477–80.Google Scholar
  4. 4.
    Hakimi SL. On the realizability of a set of integers as degrees of the vertices of a graph. SIAM J Appl Math. 1962;10:496–506.CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Erdὅs P, Gallai T. Graphs with prescribed degree of vertices. (Hungarian) Math Lapok. 1960;11:264–74.Google Scholar
  6. 6.
    Ryser HJ. Combinatorial properties of matrices of zeros and ones. Can J Math. 1957;9:371–7.CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Berge C. Graphs and hypergraphs. Amsterdam: North Holland and New York: American Elsevier; 1973.Google Scholar
  8. 8.
    Fulkerson DR, Hofman AJ, McAndrew MH. Some properties of graphs with multiple edges. Can J Math. 1965;17:166–77.CrossRefzbMATHGoogle Scholar
  9. 9.
    Bollobảs B. Extremal graph theory. New York: Acedemic Press; 1978.Google Scholar
  10. 10.
    Grὔnbaum B. Graphs and complexes. Report of the university of Washington, Seattle, Math. 572B, (1969) (private communication).Google Scholar
  11. 11.
    Hässelbarth W. Die Verzweighteit von Graphen. Match. 1984;16:3–17.zbMATHGoogle Scholar
  12. 12.
    Sierksma G, Hoogeveen H. Seven criteria for integer sequences being graphic. J Graph Theory. 1991;15(2):223–31.CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Kleitman DJ, Wang DL. Algorithms for constructing graphs and digraphs with given valences and factors. Discrete Math. 1973;6:79–88.CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Czajkowski, Eggleton RB. Graphic sequences and graphic polynomials: a report. Infinite Finite Sets. 1973;1:385–92.Google Scholar
  15. 15.
    Tripathi A, Vijay S. A note on theorem on Erdὅs and Gallai. Discrete Math. 2003;265:417–20.CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Dahl G, Flatberg T. A remark concerning graphical sequences. Discrete Math. 2005;304:62–4.CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Tripathi A, Taygi H. A simple criterion on degree sequences of graphs. Discrete Appl Math. 2008;156:3513–7.CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Boukerche A, Tropper C. A distributed graph algorithm for the detection of local cycles and knots. IEEE Trans Parallel Distrib Syst. 1998;9:748–57.Google Scholar
  19. 19.
    Manivannan D, Singhal M. An efficient distributed algorithm for detection of knots and cycles in a distributed graph. IEEE Trans Parallel Distrib Syst. 2003;14:961–72.Google Scholar

Copyright information

© Springer India 2016

Authors and Affiliations

  • Prantik Biswas
    • 1
    Email author
  • Shahin Shabnam
    • 2
  • Abhisek Paul
    • 1
  • Paritosh Bhattacharya
    • 1
  1. 1.Department of Computer Science and EngineeringNational Institute of TechnologyAgartalaIndia
  2. 2.Department of Computer ScienceAssam UniversitySilcharIndia

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