Degree Associated Edge Reconstruction Number

  • S. Monikandan
  • S. Sundar Raj
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7643)


An edge-deleted subgraph of a graph G is called an ecard of G. An ecard of G with which the degree of the deleted edge is also given is called a degree associated ecard (or da-ecard) of G. The edeck (da-edeck) of a graph G is its collection of ecards (da-ecards). The degree associated edge reconstruction number, dern(G), of a graph G is the size of the smallest collection of ecards of G uniquely determines G. The adversary degree associated edge reconstruction number, adern(G), of a graph G is the minimum number k such that every collection of k da-ecards of G uniquely determines G. We prove that dern(G)= adern(G)=1 for any regular graph G or any bidegreed graph G with exactly one vertex of different degree, which differs by at least three. We determine dern and adern for all complete bipartite graphs except K 1,3. We also prove that dern(G)≤ 2 and adern(G)≤ 3 for any complete 3-partite graph G with n vertices in which all partite sets are equal in size as possible and a few other results.


reconstruction number edge reconstruction number card dacard 


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  1. 1.
    Harary, F.: Graph Theory. Addison Wesley, Mass. (1969)Google Scholar
  2. 2.
    Harary, F.: On the reconstruction of a graph from a collection of subgraphs. In: Fieldler, M. (ed.) Theory of Graphs and its Applications, pp. 47–52. Academic Press, New York (1964)Google Scholar
  3. 3.
    Harary, F., Plantholt, M.: The graph reconstruction number. J. Graph Theory 9, 451–454 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Myrvold, W.J.: The ally and adversary reconstruction problems. Ph.D. Thesis, University of Waterloo (1988)Google Scholar
  5. 5.
    Myrvold, W.J.: The ally-reconstruction number of a disconnected graph. Ars Combin. 28, 123–127 (1989)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ramachandran, S.: Degree associated reconstruction number of graphs and digraphs. Mano. Int. J. Mathematical Sciences 1, 41–53 (2000)Google Scholar
  7. 7.
    Ramachandran, S.: The Reconstruction number for Ulam’s Conjecture. Ars Combin. 78, 289–296 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Barrus, M.D., West, D.B.: Degree-associated reconstruction number of graphs. Discrete Math. 310, 2600–2612 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Molina, R.: The Edge Reconstruction Number of a Disconnected Graph. J. Graph Theory 19(3), 375–384 (1995)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • S. Monikandan
    • 1
  • S. Sundar Raj
    • 2
  1. 1.Department of MathematicsManonmaniam Sundaranar UniversityTirunelveliIndia
  2. 2.Department of MathematicsVivekananda CollegeKanyakumariIndia

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