Graphs and Combinatorics

, Volume 3, Issue 1, pp 285–291

# On thel-connectivity of a graph

• Ortrud R. Oellermann
Article

## Abstract

For an integerl ≥ 2, thel-connectivity of a graphG is the minimum number of vertices whose removal fromG produces a disconnected graph with at leastl components or a graph with fewer thanl vertices. A graphG is (n, l)-connected if itsl-connectivity is at leastn. Several sufficient conditions for a graph to be (n, l)-connected are established. IfS is a set ofl(≥ 3) vertices of a graphG, then anS-path ofG is a path between distinct vertices ofS that contains no other vertices ofS. TwoS-paths are said to be internally disjoint if they have no vertices in common, except possibly end-vertices. For a given setS ofl ≥ 2 vertices of a graphG, a sufficient condition forG to contain at leastn internally disjointS-paths, each of length at most 2, is established.

### Keywords

Distinct Vertex Condition forG

## Preview

### References

1. 1.
Bondy, J.A.: Properties of graphs with constraints on degrees. Stud. Sci. Math. Hung.4, 473–475 (1969)Google Scholar
2. 2.
Chartrand, G., Kapoor, S.F., Kronk, H.V.: A sufficient condition forn-connectedness of graphs. Mathematika15, 51–52 (1968)Google Scholar
3. 3.
Chartrand, G., Kapoor, S.F., Lesniak, L., Lick, D.R.: Generalized connectivity in graphs. Bull. Bombay Math. Colloq.2, 1–6 (1984)Google Scholar
4. 4.
Chartrand, G., Lesniak, L.: Graphs & Digraphs, Second Edition. Monterey: Wadsworth & Brooks/Cole 1986Google Scholar
5. 5.
Hedman, B.: A sufficient condition forn-short-connectedness. Math. Mag.47, 156–157 (1974)Google Scholar
6. 6.
7. 7.
Menger, K.: Zur allgemeinen Kurventheorie. Fund. Math.10, 96–115 (1927)Google Scholar
8. 8.
Whitney, H.: Congruent graphs and the connectivity of graphs. Amer. J. Math.54, 150–168 (1932)Google Scholar