Reconstruction of a graph from 2–vicinities of its vertices1,2

https://doi.org/10.1016/j.endm.2005.07.065Get rights and content

Abstract

We prove that a connected graph of diameter at least 4 and of girth 7 or more (in particular, a tree) can be exactly reconstructed from metric balls of radius 2 of all its vertices. On the other hand, there exist graphs of diameter 3 and of girth 6 which are not reconstructible. This new graph theory problem is motivated by reconstruction of chemical compounds.

References (16)

  • V.I. Levenshtein

    Efficient reconstruction of sequences from their subsequences or supersequences

    Journal of Combin. Theory, Ser. A

    (2001)
  • A.V. Aho et al.

    The Design and Analysis of Computer Algorithms

    (1976)
  • J.A. Bondy

    A graph reconstructor's manual, Surveys in combinatorics

  • K.A. Blinov et al.

    An expert system for automated structure elucidation utilizing proton–proton, carbon–proton and nitrogen–hydrogen 2D NMR correlations

    Fresenius J. Anal. Chem

    (2001)
  • P. Erdös et al.

    Gráfok elöirt fokú pontokkal

    Mat. lapok

    (1960)
  • F. Harary

    On the reconstruction of a graph from a collection of subgraphs

  • P.J. Kelly, On isometric transformations, Ph.D. Thesis, University of Winconsin,...
  • P.J. Kelly

    A congruence theorem for trees

    Pacific J. Math.

    (1957)
There are more references available in the full text version of this article.

Cited by (1)

1

The work was supported by the RFBR grant 04-01-00122.

2

Partially supported by the RFBR grants 03-01-00796, 04-01-00122.

View full text