1. Introduction
Graph theory has seen an explosive growth due to interaction with areas like computer science, operation research, etc. In particular, it has become one of the most powerful mathematical tools in the analysis and study of the architecture of a network. The most common networks are telecommunication networks, computer networks, road and rail networks and other logistic networks [
1]. In a communication network, the measures of vulnerability are essential to guide the designers in choosing a suitable network topology. They have an impact on solving difficult optimization problems for networks [
2,
3].
The graph vulnerability relates to the study of a graph when some of its elements (vertices or edges) are removed. The measures of graph vulnerability are usually invariants that measure how a deletion of one or more network elements changes properties of the network [
4]. In the literature, various measures have been defined to measure the robustness of a network and a variety of graph theoretic parameters have been used to derive formulas to calculate network vulnerability. The best known measure of reliability of a graph is its connectivity. The connectivity is defined to be the minimum number of vertices whose deletion results in a disconnected or trivial graph [
5].
The connectivity of a graph
G is denoted by
and it is defined as follows:
where
is the number of components of the graph
.
The toughness [
6], the integrity [
7], the domination number [
8], the bondage number [
9,
10], the edge eccentric connectivity number [
11], etc., have been proposed for measuring the vulnerability of networks. Recently, some average vulnerability parameters like the average lower independence number [
12,
13], the average lower domination number [
13,
14,
15,
16,
17], the average connectivity number [
18], the average lower connectivity number [
19] and the average lower bondage number [
4] have been defined.
Let
be a simple undirected graph of order
n. We begin by recalling some standard definitions that we need throughout this paper. For any vertex
, the open neighborhood of
v is
and closed neighborhood of
v is
. The degree of vertex
v in
G denoted by
, that is, the size of its open neighborhood [
8]. The minimum degree of graph
G is denoted by
. A set
is a dominating set if every vertex in
is adjacent to at least one vertex in
S. The minimum cardinality taken over all dominating sets of
G is called the domination number of
G and denoted by
[
8]. Another domination concept is
2-domination number. A
2-dominating set of a graph
G is a set
of vertices of graph
G such that every vertex of
has at least two neighbors in
D. The
2-domination number of a graph
G, denoted by
, is the minimum cardinality of a
2-dominating set of the graph
G [
8,
20,
21,
22].
In 2004, Henning introduced the concept of average domination and average independence in [
13]. Moreover, the average lower domination and average lower independence number are the theoretical vulnerability parameters for a network that modeled a graph [
12,
15]. The average lower domination number of a graph
G, denoted by
, is defined as follows:
where the lower domination number, denoted by
, is the minimum cardinality of a dominating set of the graph
G that contains the vertex
v [
13,
16]. In [
15], an algorithm is given for computing the average lower domination number of any graph
G.
In 2015, a new graph theoretical parameter namely the average lower 2-domination number was defined in [
23,
24]. The average lower 2-domination number of a graph
G, denoted by
, is defined as follows:
where the lower 2-domination number, denoted by
, is the minimum cardinality of a dominating set of the graph
G that contains the vertex
v [
23,
24].
If we think of a graph as modeling a network, then the average lower 2-domination number can be more sensitive for the vulnerability of graphs than the other known vulnerability measures of a graph [
23]. We consider two connected simple graphs
G and
H in
Figure 1, where
and
. Graphs
G and
H have not only equal the connectivity but also equal the domination number, the average lower domination number and the 2-domination number such as
,
,
and
. The results can be checked by readers. So, how can we distinguish between the graphs
G and
H?
When we compute
and
, we get
and
. So, the average lower 2-domination number may be used for distinguish between these two graphs
G and
H. Since
, we can say that the graph
H is more vulnerable than the graph
G. In other words, the graph
G is tougher than the graph
H [
23,
24].
The wheel graph has been used in different areas such as the wireless sensor networks, the vulnerability of networks, and so on. The wheel graph has many good properties. From the standpoint of the hub vertex, all elements, including vertices and edges, are in its one-hop neighborhood, which indicates that the wheel structure is fully included in the neighborhood graph of the hub vertex. Furthermore, wheel graphs are important for localizability because they are globally rigid in 2D space, which indicates an approach to identifying localizable vertices [
25]. Moreover, the wheels and various related graphs have been studied for many reasons. The gear graphs, the friendship graph, the helm graphs and the sun flower graphs are among such graphs. The definitions of these graphs will be given in
Section 3. In [
26], Aytac and Odabas compute the residual closeness for wheels and related graphs. In [
27], Javaid and Shokat give upper bounds for the cardinality of vertices in some wheel related graphs with a given partition dimension
k.
Our aim in this paper is to study a new vulnerability parameter, called the average lower 2-domination number. In
Section 2, well-known basic results are given for the average lower domination number, the average lower 2-domination number and the 2-domination number. In
Section 3, we compute the average lower 2-domination numbers of wheels and some related graphs. Finally, an algorithm is proposed for computing the 2-domination number and the average lower 2-domination numbers of any given graph in
Section 4.
3. The Average Lower 2-Domination Number of Wheels Related Graphs
In this section, we have calculated the average lower 2-domination number of wheels and related graphs such as the wheel graph , the gear graph , the friendship graph , the helm graph and the sun flower graph . Now, we recall the definitions of these graphs.
Definition 1. [
26] The wheel graph
with
n spokes is a graph that contains an
n-cycle and one additional central vertex
that is adjacent to all vertices of the cycle. Wheel graph
has
-vertices and
-edges.
Definition 2. [
12] The gear graph
is a wheel graph with a vertex added between each pair adjacent graph vertices of the outer cycle. The gear graph
has
-vertices and
-edges.
Definition 3. [
26] The friendship graph
is collection of
n triangles with a common vertex. The friendship graph
has
-vertices and
-edges.
Definition 4. [
27] The helm graph
is the graph obtained from an
n-wheel graph by adjoining a pendant edge at each vertex of the cycle. The helm graph
has
-vertices and
-edges.
Definition 5. [
27] The sun flower graph
the graph obtained from an
n-wheel graph with central vertex
and
n-cycle and additional
n vertices
where
is joined by edges to
for
where
is taken modulo
n. The sun flower graph
has
-vertices and
-edges.
We display the graphs
and
in
Figure 2.
Theorem 17. If is a wheel graph of order , where , then .
Proof. The - set of a graph , , is a set with the vertex and vertices from the set . So, . Thus, is obtained for every vertex . As a result, we get .
Remark 1. Let and be wheels graph with order 3 and 4, respectively. Then, and .
Remark 2. If is a wheel graph of order , then .
Theorem 18. If is a gear graph of order , then .
Proof. We partition the vertices of graph
into three subsets
,
and
as follows:
When the is calculated for all vertices v in the graph , each vertex satisfies one of the three cases below.
Case 1. Let be the vertex of . The center vertex is adjacent to n vertices in . Thus, all vertices of are 1-dominated. By the definition of gear graphs, the whole vertex set (or ) is taken to -set, then is obtained.
Case 2. Let be the vertex of . Clearly every vertex of the graph is 2-dominated by the vertices of . As a result, we have , where .
Case 3. Let be the vertex of . The -set including vertex is similar to -set in the Case 1. So, we have , where .
By Cases 1, 2 and 3, we have:
Theorem 19. If is a friendship graph of order , then .
Proof. By the definition of the friendship graph and 2-domination number, a -set must include the vertex whose degree is 2n. Thus, 2n-vertices are 1-dominated by the vertex . Furthermore, n-disjoint graphs are formed by these 2n-vertices in the graph . When any vertex of each graph is taken to a -set, is obtained. It is easy to see that for every vertex . Thus, we get .
Theorem 20. If is a helm graph of order , then .
Proof. Since the -set is unique in the graph , we have by the Theorem 10. As a result, is obtained.
Theorem 21. If is a sun flower graph of order , then .
Proof. The proof follows directly from the Theorem 18.
It is point out that the gear graph
is tougher than the friendship graph
and the helm graph
, where
and
. Similarly, the wheel graph
is tougher than the sun flower graph
, where
and
. Readers can see that these results are shown in
Figure 3 and
Figure 4.
4. An Algorithm for Computing the Average Lower 2-Domination Number
In this section, the algorithm in [
29] which finds the domination number and all the minimal dominating sets of a graph is improved. The improved algorithm also computes the 2-domination number and the average lower 2-domination number of a graph. The definitions used in the algorithm below are found in [
29].
positive integer
element of
array of
: real number
BEGIN
for to do
begin
;
if then end if;
if then
ELSE
for to do
begin
for to do
begin
if and and and
then end if;
end; {for k}
end; {for i}
end if;
end; {for j}
for to do
begin
;
end;
;
;
for do
;
;
for to do
begin
;
for to do
begin
if then end if;
if then end if;
if then end if;
end; {for j}
;
end; {for i}
;
END.
Example 1. Compute the 2-domination number and the average lower 2-domination number of graph
G in
Figure 5.
Firstly, we must find function
f as follows:
Then, two mathematical logic functions are used as follows:
Thus, we have
Furthermore, we have .
Clearly, the 2-domination sets and have been found by the algorithm. Thus, we get .