Cyclic Vertex Connectivity of Star Graphs

For a connected graph G, a vertex subset F ⊂ V (G) is a cyclic vertex-cut of G if G - F is disconnected and at least two of its components contain cycles. The cardinality of a minimum cyclic vertex-cut of G, denoted by κc(G), is the cyclic vertex-connectivity of G. In this paper, we show that for any integer n > 4, the n-dimensional star graph SGn has κc(SGn) = 6(n - 3).

[1]  Khaled Day,et al.  A Comparative Study of Topological Properties of Hypercubes and Star Graphs , 1994, IEEE Trans. Parallel Distributed Syst..

[2]  Dingjun Lou,et al.  Lower bound of cyclic edge connectivity for n-extendability of regular graphs , 1993, Discret. Math..

[3]  Zhao Zhang,et al.  A kind of conditional vertex connectivity of star graphs , 2009, Appl. Math. Lett..

[4]  Marc J. Lipman,et al.  Increasing the connectivity of the star graphs , 2002, Networks.

[5]  Abdol-Hossein Esfahanian,et al.  Generalized Measures of Fault Tolerance with Application to N-Cube Networks , 1989, IEEE Trans. Computers.

[6]  M. Watkins.  Connectivity of transitive graphs , 1970 .

[7]  Qi Liu,et al.  2-restricted Edge Connectivity of Vertex-transitive Graphs , 2004, Australas. J Comb..

[8]  Zhao Zhang,et al.  On cyclic edge-connectivity of transitive graphs , 2009, Discret. Math..

[9]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[10]  Shuo-Cheng Hu,et al.  Fault tolerance on star graphs , 1997, Proceedings the First Aizu International Symposium on Parallel Algorithms/Architecture Synthesis.

[11]  Dingjun Lou,et al.  On the 2-extendability of planar graphs , 1991, Discret. Math..

[12]  Neil Robertson,et al.  Minimal cyclic-4-connected graphs , 1984 .

[13]  Roman Nedela,et al.  Atoms of cyclic connectivity in cubic graphs , 1995 .

[14]  Sheldon B. Akers,et al.  A Group-Theoretic Model for Symmetric Interconnection Networks , 1989, IEEE Trans. Computers.

[15]  Cun-Quan Zhang.  Integer Flows and Cycle Covers of Graphs , 1997 .

[16]  Marie-Claude Heydemann,et al.  Cayley graphs and interconnection networks , 1997 .

[17]  Xu Jun-ming.  On Restricted Edge-Connectivity of Vertex-Transitive Graphs , 2004 .

[18]  Sheldon B. Akers,et al.  The Star Graph: An Attractive Alternative to the n-Cube , 1994, ICPP.

[19]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[20]  Shahram Latifi,et al.  Conditional Connectivity Measures for Large Multiprocessor Systems , 1994, IEEE Trans. Computers.

[21]  Tait.  4. On the Colouring of Maps. , 1880 .

[22]  M. Plummer.  On the cyclic connectivity of planar graphs , 1972 .