## Abstract A result of G. Chartrand, A. Kaugars, and D. R. Lick [Proc Amer Math Soc 32 (1972), 63β68] says that every finite, kβconnected graph __G__ of minimum degree at least β3__k__/2β contains a vertex __x__ such that __G__β__x__ is still __k__βconnected. We generalize this result by proving t
Connectivity keeping trees in k-connected graphs
β Scribed by W. Mader
- Publisher
- John Wiley and Sons
- Year
- 2011
- Tongue
- English
- Weight
- 86 KB
- Volume
- 69
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
β¦ Synopsis
We show that one can choose the minimum degree of a k-connected graph G large enough (independent of the vertex number of G) such that G contains a copy T of a prescribed tree with the property that G -V (T ) remains k-connected.
π SIMILAR VOLUMES
Let T(G) be the tree graph of a graph G with cycle rank r. Then K ( T ( G ) ) 3 m ( G ) -r, where K(T(G)) and m(G) denote the connectivity of T ( G ) and the length of a minimum cycle basis for G, respectively. Moreover, the lower bound of m ( G ) -r is best possible.
## Abstract For a graph __G__, a subset __S__ of __V__(__G__) is called a shredder if __G__βββ__S__ consists of three or more components. We show that if __k__ β₯ 4 and __G__ is a __k__βconnected graph, then the number of shredders of cardinality __k__ of __G__ is less than 2|__V__(__G__)|/3 (we sho
## Abstract For a graph __G__ we define a graph __T__(__G__) whose vertices are the triangles in __G__ and two vertices of __T__(__G__) are adjacent if their corresponding triangles in __G__ share an edge. Kawarabayashi showed that if __G__ is a __k__βconnected graph and __T__(__G__) contains no ed
## Abstract We show that every __k__βconnected graph with no 3βcycle contains an edge whose contraction results in a __k__βconnected graph and use this to prove that every (__k__ + 3)βconnected graph contains a cycle whose deletion results in a __k__βconnected graph. This settles a problem of L. Lo