The reconstruction problem for certain infinite graphs

Abstract

We are concerned with the notion of the degree-type (D iG )i∈ω of a graphG, whereD iG is defined to be the number of vertices inG with degreei. In the first section the following results are proven:

  1. i)

    IfG is a connected, locally finite, countably infinite graph such that there exists ani so thatD iG andD i+1 G are both finite and different from 0, thenG is reconstructible.

  2. ii)

    Locally finite, countably infinite graphsG, for which infinitely manyD iG are different from 0 but only finitely manyD iG are infinite, are reconstructible.

In the second section we give some results about the reconstructibility of certain locally finite countably infinite interval graphs and show that a reconstruction of a planar, infinite graph has to be planar too.

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References

  1. [1]

    T. Andreae, On reconstruction of locally finite trees,Preprint; Freie Universität Berlin 1979.

  2. [2]

    J. A. Bondy andR. L. Hemminger, Reconstructing infinite graphs,Pacific Journ. of Math. 52 (1974), 331–339.

    MATH  MathSciNet  Google Scholar 

  3. [3]

    M. von Rimscha, Reconstructibility and perfect graphs,Accepted for publication in: Discrete Mathematics.

  4. [4]

    C. Thomassen, Infinite graphs,To appear in: Selected topics in graph theory II, (L. W. Beineke and R. J. Wilson, eds.); Academic Press.

  5. [5]

    K. Wagner, Falsplättbare Graphen,Journal of Comb. Theory 3 (1967), 326–365.

    MATH  Google Scholar 

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v. Rimscha, M. The reconstruction problem for certain infinite graphs. Combinatorica 4, 339–343 (1984). https://doi.org/10.1007/BF02579145

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AMS subject classification (1980)

  • 05 C 60
  • 04 A 20