1. Introduction
In this paper we shall use the terminology of [1]. Let be a simple graph with vertex set and edge set. Let,. For every, the open neighborhood of, denoted by, is a set and the closed neighborhood of, denoted by, is a set. We write for the degree of a vertex and the maximum and minimum degree of are denoted by and, respectively. For every, the edge-closed neighborhood of, denoted by, is
.
Many domination parameters in graphs has been studied richly [2-4] A function is a signed dominating function if for every vertex
,.
The weight of is the sum of the function values of all vertices in. The signed domination number of is the minimum weight of signed dominating functions on. This concept was introduced by Dunbar et al. [5] and has been studied by several authors [6-9]. As an extension of the signed domination, we give the definition of the reverse total signed vertex domination in a graph.
Definition 1. Let be a simple graph. A reverse total signed vertex dominating function of is a function such that
for all. The reverse total signed vertex domination number of, denoted by, is the maximum weight of a reverse total signed vertex dominating function of. A reverse total signed vertex dominating function is called a -function of if .
2. Properties of Reverse Total Signed Vertex Domination
Proposition 1 For any graph,
.
Proof. Let be a -function of. Then
.
Let
,
,
,
.
Then
.
Therefore.
Propositon 2 For any graph,.
Proof. Let be a -function of. Then for every, and we have
Thus.
Propositon 3 For any graph,.
Proof. Let be a -function of., , and are defined as Proposition 2. Then
.
We define two induced graphs and of as follows:
, ,.
Then for every,
and. For every, we have
and. Thus
Therefore
Since
we have. Therefore.
Propositon 4 For any star,.
Proof. Let be a -function. Let
,
,
where is the center of. Since for every, , we have
.
On the other hand, consider the function
such that
,.
Then is a reverse total signed vertex dominating function on and
.
Thus, which implies that .
Propositon 5 For any circle,.
Proof. Let be a -function of. Let
,.
Since for every, , we have
.
Thus
.
Therefore.
On the other hand, consider the mapping
such that
,.
Then is a reverse total signed vertex dominating function on and. Therefore
which implies.
Propositon 6 For any complete bipartite graph,.
Proof. Letbe a -function. Let
, ,
and
.
Since for every, , we have . Therefore
.
On the other hand, consider the mapping
such that, for,
for and. Then is a reverse total signed vertex dominating function on
and. Thereforewhich implies.
3. Acknowledgements
This work was supported by the Natural Science Foundation of Hebei Province (A2012408002), the Educational Commission of Hebei Province (ZH2011122, Z2011157) and Langfang Teachers College (LSZQ201106).