Global neighbourhood domination

A subset of vertices of a graph is called a global neighbourhood dominating set(gnd set) if is a dominating set for both and , where is the neighbourhood graph of . The global neighbourhood domination number(gnd number) is the minimum cardinality of a global neighbourhood dominating set of and is denoted by ( ). In this paper sharp bounds for , are supplied for graphs whose girth is greater than three. Exact values of this number for paths and cycles are presented as well. The characterization result for a subset of the vertex set of to be a global neighbourhood dominating set for is given and also characterized the graphs of order having gnd numbers 1 2 1 2 . Subject Classification : 05C69.


Introduction & Preliminaries
Domination is an active subject in graph theory, and has numerous applications to distributed computing, the web graph and adhoc networks.For a comprehensive introduction to theoretical and applied facets of domination in graphs the reader is directed to the book [4].
A set of vertices is called a dominating set of if each vertex not in is joined to some vertex in .The domination number ( ) is the minimum cardinality of the dominating set of [4].
Many variants of the domination number have been studied.For instance a dominating set of a graph is called a restrained dominating set if every vertex in is adjacent to a vertex in as well as another vertex in .The restrained domination number of , denoted by ( ) is the smallest cardinality of the restrained dominating set of [3].A set is called a global dominating set of if is a dominating set of both and its complement .The global domination number of , denoted by ( ) is the smallest cardinality of the global dominating set of [6].A dominating set of connected graph is called a connected dominating setof if the induced subgraph is connected.The connected domination number of , denoted by ( ) is the smallest cardinality of the connected dominating set of [7].A dominating set of connected graph is called an independent dominating set of if the induced subgraph is a null graph [4].be a connected graph, then the Neighbourhood Graph of is denoted by ( ) ( ) and it has the same vertex set as that of and edge set being { ( ) ( ) ( )} [2].Recently we have introduced a new type of graph known as semi complete graph.Let be a connected graph, then is said to be semi complete if any pair of vertices in have a common neighbour.The necessary and su cient condition for a connected graph to be semi complete is any pair of vertices lie on the same triangle or lie on two di erent triangles having a common vertex [5].
In the present paper, we introduce a new graph parameter, the global neighbourhood domination number, for a connected graph .We call ( ) a global neighbourhood dominating set (gnd -set) of if is a dominating set for both .The global neighbourhood domination number is the minimum cardinality of a global neighbourhood dominating set of and is denoted by ( ).
Example.Suppose is a graph representing a network of roads linking various locations.Some essential goods are being supplied to these locations from supplying stations.It may happen that these links(edges of ) may be broken for some reason or the other.So we have to think of maintaining the supply of goods to various locations uninterrupted through secret links(edges in the neighbourhood graph of ).As the neighbourhood graph of is a spanning subgraph of , the construction (maintainance) cost of secret links can be minimized, when compared with the complementary graph of .The global neighbourhood domination number will be the minimum number of supplying stations needed to accomplish the task of supplying the goods uninterruptedly.
All graphs considered in this paper are simple, finite, undirected and connected.For all graph theoretic terminology not defined here, the reader is referred to [1].
In section [2], sharp bounds for are supplied for the graphs whose girth is greater than three.In section [3], we have given a characterization result for a proper subset of the vertex set of to be a gnd -set of and also characterized the graphs whose gnd -numbers are 1 2 1 2.

Bounds for the global neighbourhood domination number
In this section, we obtain some bounds for the gnd -numbers of graphs whose girth is greater than three.Proof: Let be a minimum gnd -set of .By hypothesis every vertex in is non adjacent with atleast one vertex in .Otherwise we get a contradiction to that is a gnd -set for .
Let 1 2 ( ) be the neighbours of in .Then ( )} is a gnd -set of and its cardinality is ( ) + 1.
From ( 1) and ( 2) Furthermore the lower bound is attained in the case of 4 and upper bound is attained in the case of 3 .Hence the bounds are sharp.

Note:
The upper bound holds good for any connected graph .be a connected graph and be a minimum dominating set of .If there is a vertex in such that is adjacent to all the vertices in , then ( ) 1 + ( ).
Proof: Assume that ( ) for some .The proof follows from the fact that S { } is a gnd -set of .
Theorem 2.3.be a minimum dominating set of .Then ( ) = 1 + ( ) i there is a vertex in satisfying: (i) ( ) , each of the vertices in ( ) is isolated in .
Proof: By hypothesis, every gnd -set is a global dominating set in .Hence ( ) ( ).
Note: Under the hypothesis given in the Theorem(2.10)and Theorem(2.9),we Theorem 2.6.be a connected graph.Then = i (i) Each edge in lies on 3 or 5 .
(ii) There is no path of length four between any pair of non adjacent vertices in .
(i) Let 1 2 be an arbitrary edge in , then by our assumption 1 2 is an edge in .Suppose 1 2 . Since 1 2 ( ), 1 2 lies on a cycle 3 in .Suppose 1 2 . Since 1 2 , there is a 3 in ( ) such that is a path in .This implies 1 3 3 2 ( ).So there is a path of length four from 1 to 2 in .Thus S { 1 2 } is a 5 -cycle in .Therefore 1 2 lies on 5 .
Hence each edge in lies on 3 or 5 .
(ii) If there is a path of length four between any pair of non adjacent vertices in , then there is an edge in which is not in .Hence 6 = , which is a contradiction.Assume that the converse holds.Let 1 2 be an arbitrary edge in .Then by (i) of our assumption 1 2 lies on 3 or 5 .In either case 1 2 is an edge in .Hence .

Characterization and Other Relevant Results.
In this section we have given the characterization for a proper subset of the vertex set of a graph to be a gnd -set.).
So in either case 1 lies on the edge whose end points are totally dominated by the vertices in .
Assume that the converse holds.

Theorem 2 . 1 .
If is a triangle free graph, then

is obtained from 3 or 4
by adding zero or more leaves to the stems of the path.Theorem 2.2.
Theorem 3.1.(Characterization Result) be a connected graph. is a gnd -set of i each vertex in lies on an edge whose end points are totally dominated by the vertices in .If 4 6 = 2 , then 1 lies on the edge 1 4 , where 1 is dominated by 2 and 4 is dominated by 3 ( 2 3