Improved diameter bounds for altered graphs
We consider the following problem: Given positive integers k and D, what is the maximum diameter of the graph obtained by deleting k edges from a graph G with diameter D, assuming that the resulting graph is still connected. For undirected graphs G we prove an upper bound of (k+1)D and a lower bound of (k+1)D-k for even D and of (k+1)D-2k+2 for odd D≥3. For directed graphs G, the bounds depend strongly on D: for D=1 and D=2 we derive exact bounds of θ (√k) and of 2k+2, respectively, while for D≥3 the resulting diameter is in general unbounded in terms of k and D.
KeywordsShort Path Directed Graph Maximum Diameter Undirected Graph Diameter Increase
Unable to display preview. Download preview PDF.
- Chung, F.R.K., Diameters of communication networks, in: M. Anshel and W. Gewirtz (eds.), Mathematics of information processing, Proc. Symposia in Applied Math., vol 34, American Math. Soc., Providence, RI, 1986, pp. 1–18.Google Scholar
- Chung, F.R.K., and M.R. Garey, Diameter bounds for altered graphs, J. Graph Theory 8 (1984) 511–534.Google Scholar
- Ghouila-Houri, A., Un résultat relatif á la notion de diamètre, C.R. Acad. Sci. de Paris 250 (1960) 4254–4256.Google Scholar
- Plesnik, J., Note on diametrically critical graphs, Recent Advances in Graph Theory, Proc. 2nd Czechoslovak Symp. (Prague 1974), Academia, Prague (1975) 455–465.Google Scholar
- Schoone, A.A., H.L. Bodlaender and J. van Leeuwen, Diameter increase caused by edge deletion, J. of Graph Theory, (to appear).Google Scholar