Abstract
In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for g(n, l), the biggest number k guaranteeing that there exist l graphs on n vertices, each two having edit distance at least k. By edit distance of two graphs G, F we mean the number of edges needed to be added to or deleted from graph G to obtain graph F. This new extremal number g(n, l) is closely linked to the edit distance of graphs. Using probabilistic methods we show that g(n, l) is close to \(\frac{1} {2}\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\) for small values of l > 2. We also present some exact values for small n and lower bounds for very large l close to the number of non-isomorphic graphs of n vertices.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Alon, A. Shapira, B. Sudakov: Additive approximation for edge-deletion problems. Ann. Math. (2) 170 (2009), 371–411.
N. Alon, J. H. Spencer: The Probabilistic Method. Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Hoboken, 2008.
N. Alon, U. Stav: What is the furthest graph from a hereditary property? Random Struct. Algorithms 33 (2008), 87–104.
M. Axenovich, A. Kézdy, R. Martin: On the editing distance of graphs. J. Graph Theory 58 (2008), 123–138.
J. Balogh, R. Martin: Edit distance and its computation. Electron. J. Comb. (electronic only) 15 (2008), Research Paper R20, 27 pages.
D. Chen, O. Eulenstein, D. Fernández-Baca, M. Sanderson: Supertrees by flipping. Computing and Combinatorics (O. H. Ibarra et al., eds.). Proc. of the 8th Annual International Conf., Singapore, 2002, Lecture Notes in Comput. Sci. 2387, Springer, Berlin, 2002, pp. 391–400.
F. R. K. Chung, P. Erdős, R. L. Graham: Minimal decompositions of graphs into mutually isomorphic subgraphs. Combinatorica 1 (1981), 13–24.
P. O. deWet: Constructing a large number of nonisomorphic graphs of order n. Morehead Electronic Journal of Applicable Mathematics 1 (2001), 2 pages.
T. Dzido, K. Krzywdziński: On a local similarity of graphs. Discrete Math. 338 (2015), 983–989.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by the Polish National Science Centre grant number 2011/02/A/ST6/00201.
Rights and permissions
About this article
Cite this article
Dzido, T., Krzywdziński, K. Edit distance measure for graphs. Czech Math J 65, 829–836 (2015). https://doi.org/10.1007/s10587-015-0211-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-015-0211-4