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 ![](https://www.scirp.org/html/11-1200129\ac3c96a4-8c7d-4da6-89ea-57b5a3a61f07.jpg)
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
![](https://www.scirp.org/html/11-1200129\a050bf5a-d4f1-42ec-ae2a-10f9035f3646.jpg)
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
,
![](https://www.scirp.org/html/11-1200129\c80017df-97a4-4e9c-81b7-28b47a306ae3.jpg)
and
. For every
, we have
![](https://www.scirp.org/html/11-1200129\6d19021f-630c-4edc-b469-d6b2b6f4db9b.jpg)
and
. Thus
![](https://www.scirp.org/html/11-1200129\2df47ee3-1790-4050-943d-4a9708c20152.jpg)
Therefore
![](https://www.scirp.org/html/11-1200129\7fd75af1-dcf8-4bfa-9974-4d094a93ec2b.jpg)
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
![](https://www.scirp.org/html/11-1200129\73fa598e-7ab9-4056-bf6b-383f030ab738.jpg)
such that
![](https://www.scirp.org/html/11-1200129\aeeb6093-0d70-4020-89ed-7bb065faf3a3.jpg)
,![](https://www.scirp.org/html/11-1200129\2aa19bae-9fe8-4488-bb6a-4355a0f1c1e0.jpg)
.
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
![](https://www.scirp.org/html/11-1200129\2262ff1c-42e8-4aca-8143-4d20ccaf83a8.jpg)
such that
![](https://www.scirp.org/html/11-1200129\b9ca98fa-ddc9-4cc9-9d5a-b7c25d543ec9.jpg)
,![](https://www.scirp.org/html/11-1200129\3d699fd6-de6d-4029-9752-e6eb55627b6b.jpg)
.
Then
is a reverse total signed vertex dominating function on
and
. Therefore
which implies
.
Propositon 6 For any complete bipartite graph![](https://www.scirp.org/html/11-1200129\82bdb2bb-671d-48ed-a5f3-7e2cefc13b46.jpg)
,
.
Proof. Let
be a
-function. Let
,
,
![](https://www.scirp.org/html/11-1200129\0522ab3d-b318-4e9e-ac02-e3ba3e1ada52.jpg)
and
.
Since for every
,
, we have
. Therefore
.
On the other hand, consider the mapping
![](https://www.scirp.org/html/11-1200129\d91141f7-49bd-47ce-94e3-a069b19b4eb8.jpg)
such that
,
for
,
for
and
. Then
is a reverse total signed vertex dominating function on ![](https://www.scirp.org/html/11-1200129\58bd211e-6fb6-456d-906a-22ff42a370f4.jpg)
and
. Therefore
which 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).