Excellent Domination Subdivision Stable Graphs
About this article
Abstract
A set of vertices D in a graph G = ( V, E ) is a dominating set if every vertex of V – D is adjacent to some vertex of D. If D has the smallest possible cardinality of any dominating set of G, then D is called a minimum dominating set — abbreviated MDS. A graph G is said to be excellent if given any vertex v then there is a g - set of G containing v. An excellent graph G is said to be very excellent ( VE ), if there is a g - set D of G such that to each vertex u Î V – D $ a vertex v Î D such that D – { v } È { u } is a g - set of G. In this paper we have proved that very excellent trees are subdivision stable. We also have provided a method of generating an excellent subdivision stable graph from a non - excellent subdivision stable graph.
References
[1] T. W. Haynes, S. T. Hedetniemi, P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, ( 1998 ).
[2] K. Karthika, Domination Dot Stable – Domatic dot stable – Domination Subdivision Stable graphs, M. Phil thesis, VIT University, Vellore, India, (2011).
[3] N. Sridharan, M. Yamuna, Very excellent graphs and Rigid very excellent graphs, AKCE J. Graph. Combin., ( 2007 ), Vol – 4, pp. 211 – 221.
[4] N. Sridharan, M. Yamuna, A note on excellent graphs, Ars. Combin, ( 2006 ), Vol – 78 , pp. 267 – 276.
[5] D. B. West, Introduction to Graph Theory, second ed., Prentice-Hall, Englewood Cliffs, NJ (2001).
View more references (2)
[6] M. Yamuna, Excellent – Just Excellent – Very Excellent graphs, Ph. D thesis, Alagappa University, Karaikudi, India ( 2003 ).
[7] M. Yamuna, K. Karthika, Domination Subdivision Stable Graphs, International Journal of Mathematical Archive ( 2012 ) - 3(4), Page: 1467-1471.