The Average Lower 2-Domination Number of Wheels Related Graphs and an Algorithm

The problem of quantifying the vulnerability of graphs has received much attention nowadays, especially in the field of computer or communication networks. In a communication network, the vulnerability measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. If we think of a graph as modeling a network, the average lower 2-domination number of a graph is a measure of the graph vulnerability and it is defined by 2 2 ( ) 1 ( ) ( ) ( ) av v v V G G G V G ∈ γ = γ  , where the lower 2-domination number, denoted by 2 ( ) v G γ , of the graph G relative to v is the minimum cardinality of 2-domination set in G that contains the vertex v. In this paper, the average lower 2-domination number of wheels and some related networks namely gear graph, friendship graph, helm graph and sun flower graph are calculated. Then, we offer an algorithm for computing the 2-domination number and the average lower 2-domination number of any graph G.


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 kpGq and it is defined as follows: kpGq " min t|S| : S Ă V and wpG ´Sq ą 1u where wpG ´Sq is the number of components of the graph G ´S.
Let G " pVpGq, EpGqq be a simple undirected graph of order n.We begin by recalling some standard definitions that we need throughout this paper.For any vertex v P VpGq, the open neighborhood of v is N G pvq " tu P V| uv P EpGqu and closed neighborhood of v is N G rvs " N G pvq Y tvu.The degree of vertex v in G denoted by d G pvq, that is, the size of its open neighborhood [8].The minimum degree of graph G is denoted by δpGq.A set S Ď VpGq is a dominating set if every vertex in VpGq ´S 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 γpGq [8].Another domination concept is 2-domination number.A 2-dominating set of a graph G is a set D Ď VpGq of vertices of graph G such that every vertex of VpGq ´D has at least two neighbors in D. The 2-domination number of a graph G, denoted by γ 2 pGq, 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 γ av pGq, is defined as follows: where the lower domination number, denoted by γ v pGq, 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 γ 2av pGq, is defined as follows: where the lower 2-domination number, denoted by γ 2v pGq, 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 |VpGq| " |VpHq| " 10 and |EpGq| " |EpHq| " 17.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 kpGq " kpHq " 1, γpGq " γpHq " 1, γ av pGq " γ av pHq " 19{5 and γ 2 pGq " γ 2 pHq " 5.The results can be checked by readers.So, how can we distinguish between the graphs G and H?
When we compute γ 2av pHq and γ 2av pGq, we get γ 2av pHq " 51{10 " 5.1 and γ 2av pGq " 50{10 " 5. So, the average lower 2-domination number may be used for distinguish between these two graphs G and H. Since γ 2av pGq ă γ 2av pHq, 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 2domination 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 2domination numbers of any given graph in Section 4.

Basic Results
In this section, well known basic results are given with regard to the average lower domination number, the average lower 2-domination number and the 2-domination number.

Theorem 1. [13] Let G be any graph of order n with the domination number
Theorem 5. [13] If n K is a complete graph of order n, then ( ) 1 . 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.

Basic Results
In this section, well known basic results are given with regard to the average lower domination number, the average lower 2-domination number and the 2-domination number.Theorem 1. [13] Let G be any graph of order n with the domination number γpGq, then γ av pGq ď γpGq `1 ´γpGq n with equality if and only if G has a unique γpGq-set.
Theorem 3. [13] If P n is a path graph of order n, then Theorem 4. [13] If C n is a cycle graph of order n, then γ av pC n q " 2.
Theorem 5. [13] If K n is a complete graph of order n, then γ av pK n q " 1.
Theorem 13. [23] If P n is a path graph of order n, then , I f n is even.
Theorem 14. [23] If C n is a cycle graph of order n, then γ 2av pC n q " tpn `1q{2u.
Theorem 15. [23] If K n is a complete graph of order n, then γ av pK n q " 2.

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 W n , the gear graph G n , the friendship graph f n , the helm graph H n and the sun flower graph S f n .Now, we recall the definitions of these graphs.Definition 1. [26] The wheel graph W n with n spokes is a graph that contains an n-cycle and one additional central vertex v c that is adjacent to all vertices of the cycle.Wheel graph W n has pn `1q-vertices and 2n-edges.Definition 2. [12] The gear graph G n is a wheel graph with a vertex added between each pair adjacent graph vertices of the outer cycle.The gear graph G n has p2n `1q-vertices and 3n-edges.Definition 3. [26] The friendship graph f n is collection of n triangles with a common vertex.The friendship graph f n has p2n `1q-vertices and 3n-edges.Definition 4. [27] The helm graph H n is the graph obtained from an n-wheel graph by adjoining a pendant edge at each vertex of the cycle.The helm graph H n has p2n `1q-vertices and 3n-edges.Definition 5. [27] The sun flower graph S f n the graph obtained from an n-wheel graph with central vertex v c and n-cycle v 0 , v 1 , v 2 , . . ., v n´1 and additional n vertices w 0 , w 1 , w 2 , . . ., w n´1 where w i is joined by edges to v i , v i`1 for i P t0, 1, . . ., n ´1u where pi `1q is taken modulo n.The sun flower graph S f n has p2n `1q-vertices and 4n-edges.
We display the graphs W 4 , G 4 , f 4 , H 4 and S f 4 in Figure 2. n + -vertices and 4n -edges.
We display the graphs 4 4 4 4 , , , W G f H and 4 Sf in Figure 2.
. As a result, we get 2 ( ) 1 Proof.We partition the vertices of graph n G into three subsets 1 V , 2 V and 3 V as follows: is calculated for all vertices v in the graph n G , each vertex satisfies one of the three cases below.

Case1. Let c
v be the vertex of 1 V .The center vertex c v is adjacent to n vertices in 2 V .Thus, all vertices of 2 V are 1-dominated.By the definition of gear graphs, the whole vertex set 2 Clearly every vertex of the graph n G is 2-dominated by the vertices of 2 V .As a result, we have 2 ( ) -set in the Case 1. So, we have 2 ( ) By Cases 1, 2 and 3, we have: Theorem 17.If W n is a wheel graph of order n `1, where n ě 5, then γ 2av pW n q " 1 `rn{3s.
Proof.The γ 2 pW n q-set of a graph W n , n ě 5, is a set with the vertex v c and rn{3s vertices from the set VpW n q tv c u. So, γ 2 pW n q " 1 `rn{3s.Thus, γ 2 pW n q " 1 `rn{3s is obtained for every vertex v P VpW n q.
Theorem 18.If G n is a gear graph of order 2n `1, then γ 2av pG n q " 2n 2 `2n`1 2n`1 .Proof.We partition the vertices of graph G n into three subsets V 1 , V 2 and V 3 as follows: When the γ 2av pG n q is calculated for all vertices v in the graph G n , each vertex satisfies one of the three cases below.
Case 1.Let v c be the vertex of V 1 .The center vertex v c is adjacent to n vertices in V 2 .Thus, all vertices of V 2 are 1-dominated.By the definition of gear graphs, the whole vertex set V 2 (or V 3 ) is taken to γ 2 pG n q-set, then γ 2v c pG n q " n `1 is obtained.Case 2. Let v i be the vertex of V 2 .Clearly every vertex of the graph G n is 2-dominated by the vertices of V 2 .As a result, we have γ 2v i pG n q " n, where i P t1, 2, . . ., nu.Case 3. Let v i be the vertex of V 3 .The γ 2 pG n q-set including vertex v i is similar to γ 2 pG n q-set in the Case 1. So, we have γ 2v i pG n q " n `1, where i P tn `1, n `2, . . ., 2nu.
By Cases 1, 2 and 3, we have: Theorem 19.If f n is a friendship graph of order 2n `1, then γ 2av p f n q " n `1.
Proof.By the definition of the friendship graph and 2-domination number, a γ 2 p f n q-set must include the vertex v c whose degree is 2n.Thus, 2n-vertices are 1-dominated by the vertex v c .Furthermore, n-disjoint graphs K 2 are formed by these 2n-vertices in the graph f n z tv c u.When any vertex of each graph K 2 is taken to a γ 2 p f n q-set, γ 2 p f n q " n `1 is obtained.It is easy to see that γ 2v p f n q " n `1 for every vertex v P Vp f n q.Thus, we get γ 2av p f n q " n `1.
Theorem 21.If S f n is a sun flower graph of order 2n `1, then γ 2av pS f n q " 2n 2 `2n`1 2n`1 .Proof.The proof follows directly from the Theorem 18.
It is point out that the gear graph G n is tougher than the friendship graph f n and the helm graph H n , where |VpG n q| " |Vp f n q| " |VpH n q| and |EpG n q| " |Ep f n q| " |EpH n q|.Similarly, the wheel graph W 2n is tougher than the sun flower graph S f n , where |VpW 2n q| " |VpS f n q| and |EpW 2n q| " |EpS f n q|.Readers can see that these results are shown in Figures 3 and 4 Proof.The proof follows directly from the Theorem 18.
It is point out that the gear graph n G is tougher than the friendship graph n f and the helm graph n H , where . Similarly, the wheel is tougher than the sun flower graph n Sf , where . Readers can see that these results are shown in Figures 3 and 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].

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].  5.

Conclusions
Communication systems are often subjected to failures and attacks.A variety of measures have been proposed in the literature to quantify the robustness of networks and a number of graph theoretic parameters have been used to derive formulas for calculating network reliability.In this

Conclusions
Communication systems are often subjected to failures and attacks.A variety of measures have been proposed in the literature to quantify the robustness of networks and a number of graph theoretic parameters have been used to derive formulas for calculating network reliability.In this paper we have studied the average lower 2-domination number for graph vulnerability.The average lower 2-domination number can be more sensitive than the other measures of vulnerability like connectivity, domination number, average lower domination number and 2-domination number.We have also studied wheel graphs and wheels related graphs.Finally, an algorithm is proposed for computing the 2-domination number and the average lower 2-domination numbers of any given graph G.

Figure 1 .
Figure 1.Graphs G and H.

Figure 1 .
Figure 1.Graphs G and H.

1 .
Compute the 2-domination number and the average lower 2-domination number of graph G in Figure5.
. By the definition of the friendship graph and 2-domination number, a 2 ( ) Thus, 2n-vertices are 1-dominated by the vertex c v whose degree is 2n.
Compute the 2-domination number and the average lower 2-domination number of graph G in Figure G rv j s " 1 then D rjs Ð D rjs `vj ELSE D rjs Ð D rjs `vj for i Ð 1 to n ´1 do begin for k Ð i `1 to n do begin if "`j " ˆi˘a nd `j " ˆk˘a nd `vj E v i ˘and `vk E v j ˘‰ then D rjs Ð D rjs `vi v k end if