Some conditional vertex connectivities of complete-transposition graphs

A subset F ? V ( G ) is called an R k -vertex-cut if G - F is disconnected and each vertex u ? V ( G ) - F has at least k neighbors in G - F . The cardinality of a minimum R k -vertex-cut is the R k -vertex-connectivity of G and is denoted by ? k ( G ) . The conditional connectivity is a measure to study the structure of networks beyond connectivity. Hypercubes form the basic classes of interconnection networks. Complete transposition graphs were introduced to be competitive models of hypercubes. In this paper, we determine the numbers ? 1 and ? 2 for complete-transposition graphs, ? 1 ( CT n ) = n ( n - 1 ) - 2 , ? 2 ( CT 4 ) = 16 and ? 2 ( CT n ) = 2 n ( n - 1 ) - 10 for n ? 5 .

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

[2]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[3]  Hou Fei-fei.  One Variety Conjectures of Complete-transposition Network , 2012 .

[4]  Wei Xiong,et al.  A kind of conditional fault tolerance of alternating group graphs , 2010, Inf. Process. Lett..

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

[6]  Eddie Cheng,et al.  A kind of conditional vertex connectivity of Cayley graphs generated by 2-trees , 2011, Inf. Sci..

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

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

[9]  Jixiang Meng,et al.  Conditional connectivity of Cayley graphs generated by transposition trees , 2010, Inf. Process. Lett..

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

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

[12]  A. D. Oh,et al.  Generalized Measures of Fault Tolerance in n-Cube Networks , 1993, IEEE Trans. Parallel Distributed Syst..

[13]  Eddie Cheng,et al.  Fault Resiliency of Cayley Graphs Generated by Transpositions , 2007, Int. J. Found. Comput. Sci..

[14]  S. Lakshmivarahan,et al.  Symmetry in Interconnection Networks Based on Cayley Graphs of Permutation Groups: A Survey , 1993, Parallel Comput..

[15]  Xiaohui Huang,et al.  A kind of conditional connectivity of Cayley graphs generated by unicyclic graphs , 2013, Inf. Sci..

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

[17]  Shahram Latifi,et al.  Transposition Networks as a Class of Fault-Tolerant Robust Networks , 1996, IEEE Trans. Computers.