Vertex-Neighbor-Scattering Number of Trees

A vertex subversion strategy of a graph   = , G V E is a set of vertices   S V G  whose closed neighborhood is deleted from . The survival subgraph is denoted by G G S . We call a cut-strategy of if S G G S is disconnected, or is a clique, or is  . The vertex-neighbor-scattering number of is defined to be G       ( ) = max S V G VNS G G S   S  , where is any cut-strategy of , and S G   G G  is the number of the components of G S . It has been proved that the computing problem of this parameter is complete, so we discuss the properties of vertex-neighbor-scattering number of trees in this paper. NP 

whose closed neighborhood is deleted from .The survival subgraph is denoted by G G S .We call a cut-strategy of if

S G
G S is disconnected, or is a clique, or is  .The vertex-neighbor-scattering number of is defined to be , where is any cut-strategy of , and

Introduction
In The vertex-neighbor-scattering number (VN ) of a connected noncomplete graph is defined as , where is any cut-S strategy of , and


is the number of the components of G S .We call a . In particular, we define the vertex-neighbor-scattering number of a complete graph n K to be 1 .A comet is a graph obtained by identifying one end of a path with the center of a star The following lemmata will be used in the next section.
Theorem 1: [1] Let be a path with order

Vertex-Neighbor-Scattering Number of Trees
Theorem 4: Let be a tree with order Then T On the other hand, is connected and with order , then for its any VN T 2) We distinguish two cases to prove .

VNS T 
Case 1: .Since , there must exist a vertex such that Case 2: . Then there exist at least one vertex in , say , such that .Let be any vertex adjacent to .Then Combing Case 1 and Case 2, we have

VNS T 
Thus the proof is completed.Theorem 5: If is any integer, where , then there is a tree of order such that l T Proof.If , then .By Theorem 3, is a tree of order satisfying , the conclusion holds. ,

= 4 n
Theorem 6: 1) When , is the unique tree with order such that .
, is the unique tree with order such that Proof. 1) Let T be a tree with order such that n the unique tree in this case.
2) If and is not isomorphic to , then 7 n  We distinguish two cases.Case 1.3:The degree of , x y and all are at most .We discuss three possibilities.

VNS T P
[2] F. Li and X.

Remark 2 :
The lower and upper bound in Theorem 4 is the best possible, it can be shown by paths and stars, respectively.
If there are at least two vertices in If there is unique vertex, say x , in

Figure 1 .
Figure 1.The trees of order 6 such that VNS = 1.
[3]we discuss the properties of the vertex-neighbor-scattering number of trees in this paper.Our definitions follow[3].
[1]we introduced a new graph parameter called "vertex-neighbor-scattering number."Wemotivate the use of this parameter by viewing a graph as a model of a spy network whose vertices represent agents and whose edges represent lines of communication.If a spy is discovered, the espionage agency can no longer trust any of the spies with whom he was in direct communication.It has been shown that the parameter can measure the "neighbor" stability of graphs[1].As many graphic parameters, the computing problem of this parameter is complete [2].