ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 31(3) (2011) 475-491
DOI: 10.7151/dmgt.1559


Lynne L. Doty

Mathematics Department
Marist College
Poughkeepsie, NY 12601, USA


For the notion of neighbor-connectivity in graphs whenever a vertex is subverted the entire closed neighborhood of the vertex is deleted from the graph. The minimum number of vertices whose subversion results in an empty, complete, or disconnected subgraph is called the neighbor-connectivity of the graph. Gunther, Hartnell, and Nowakowski have shown that for any graph, neighbor-connectivity is bounded above by κ. Doty has sharpened that bound in abelian Cayley graphs to approximately [1/2]κ. The main result of this paper is the constructive development of an alternative, and often tighter, bound for abelian Cayley graphs through the use of an auxiliary graph determined by the generating set of the abelian Cayley graph.

Keywords: Cayley graphs, neighbor-connectivity bound.

2010 Mathematics Subject Classification: 05C25, 05C40.


[1] I.J. Dejter and O. Serra, Efficient dominating sets in Cayley graphs, Discrete Appl. Math. 129 (2003) 319-328, doi: 10.1016/S0166-218X(02)00573-5.
[2] L.L. Doty, A new bound for neighbor-connectivty of abelian Cayley graphs, Discrete Math. 306 (2006) 1301-1316, doi: 10.1016/j.disc.2005.09.018.
[3] L.L. Doty, R.J. Goldstone and C.L. Suffel, Cayley graphs with neighbor connectivity one, SIAM J. Discrete Math. 9 (1996) 625-642, doi: 10.1137/S0895480194265751.
[4] R.J. Goldstone, The structure of neighbor disconnected vertex transitive graphs, Discrete Math. 202 (1999) 73-100, doi: 10.1016/S0012-365X(98)00348-3.
[5] G. Gunther, Neighbour-connectivity in regular graphs, Discrete Appl. Math. 11 (1985) 233-243, doi: 10.1016/0166-218X(85)90075-7.
[6] G. Gunther and B. Hartnell, On minimizing the effects of betrayals in a resistance movement, in: Proc. 8th Manitoba Conf. on Numerical Mathematics and Computing (Winnipeg, Manitoba, Canada, 1978) 285-306.
[7] G. Gunther and B. Hartnell, Optimal k-secure graphs, Discrete Appl. Math. 2 (1980) 225-231, doi: 10.1016/0166-218X(80)90042-6.
[8] G. Gunther, B. Hartnell and R. Nowakowski, Neighbor-connected graphs and projective planes, Networks 17 (1987) 241-247, doi: 10.1002/net.3230170208.
[9] J. Huang and J.-M. Xu, The bondage numbers and efficient dominations of vertex-transitive graphs, Discrete Math. 308 (2008) 571-582, doi: 10.1016/j.disc.2007.03.027.
[10] N. Obradovic, J. Peters and G. Ruzic, Efficient domination in circulant graphs with two chord lengths, Information Processing Letters 102 (2007) 253-258, doi: 10.1016/j.ipl.2007.02.004.

Received 2 September 2009
Revised 5 May 2010
Accepted 17 May 2010