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Some Generalized Theorems on Connectivity

Published online by Cambridge University Press:  20 November 2018

R. E. Nettleton
R. E. Nettleton*
Affiliation:
The Rice Institute
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The “k-dense” subgraphs of a connected graph G are connected and contain neighbours of all but at most k-1 points. We consider necessary and sufficient conditions that a point be in Γk, the union of the minimal k-dense subgraphs. It is shown that Γk contains all the [m, k]-isthmuses” and [m, k]-articulators“— minimal subgraphs which disconnect the graph into at least k + 1 disjoint graphs—and that an [m, k]-isthmus or [m, k]-articulator of Γk disconnects G. We define “central points,” “degree” of a point, and “chromatic number” and examine the relationship of these concepts to connectivity. Many theorems contain theorems previously proved (1) as special cases.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 546 - 554
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Nettleton, R.E., Goldberg, K., and Green, M.S., Dense subgraphs and connectivity, Can. J. Math., 11 (1959), 262–268.Google Scholar
2. Harary, F. and Norman, R.Z., The dissimilarity characteristic of Husimi trees, Ann. Math., 58 (1953), 134–141.Google Scholar
3. Dirac, G.A., A theorem of Brooks R.L. and a conjecture of H. Hadwiger, Proc. Lond. Math Soc, 7 (1957), 161–195.Google Scholar
4. Dirac, G.A., The structure of k-chromatic graphs, Fund. Math., 40 (1953), 42–55.Google Scholar