Determination and Testing the Domination Numbers of Helm Graph, Web Graph and Levi Graph Using MATLAB

A set is dominating set of a graph G, if every vertex in V-S is adjacent to at least one vertex in S . The domination number denoted by is defined to be the minimum cardinality of dominating set in G. We investigate the domination numbers of Helm graph, web graph and Levi graph. Also we testing our theoretical results in computer by introduce a matlab procedure to calculate the domination numbers , dominating set S and draw this graphs that illustrated the vertices of domination this graphs. It is proved that: Determination and Testing the Domination Numbers of Helm Graph, Web ... 104

is dominating set of a graph G, if every vertex in V-S is adjacent to at least one vertex in S .The domination number denoted by is defined to be the minimum cardinality of dominating set in G.We investigate the domination numbers of Helm graph, web graph and Levi graph.Also we testing our theoretical results in computer by introduce a matlab procedure to calculate the domination numbers , dominating set S and draw this graphs that illustrated the vertices of domination this graphs.It is proved that: Throughout this paper we consider only finite, undirected and simple graph (i.e.without loops and multiple edges).Our terminology and notations will be standard except as indicated.For undefined terms see [2], [3] and [4].Let G=(V, E) be a graph, where V denotes the set of vertices in G and E denotes the set of edges in G. or equivalently every vertex in V-S is adjacent to at least one vertex in S. The domination number is the minimum cardinality of dominating set in G [1].In other words we defined the domination number of a graph G by to be the order of smallest dominating set of G.A dominating set S with is called a minimum dominating set [8]. Dominating set appear to have their origins in the game of chess where the goal is to cover or dominate various squares of a chessboard by certain chess pieces.The problem of determine domination numbers of graphs first emerged in 1862 of de Jaenisch he wanted to find the minimal number of queens on chessboard, such that every square is either occupied by a queen or can be reached by a queen with single move [4].Domination can be useful tool in many chemical structures [7] also there is many application of domination theory in wireless communication networks [5] business network and making decisions.

Definition(3):
The Levi graph can be written as where and are shown in Fig. 3

see [11]
The following theorem is useful in our work:  On domination numbers of Helm graph : Theorem(2): Proof: For let the vertices of this graph labeled by: .(See Fig. 4) We given the dominating set of this graph by a set isomorphic to S Where The minimality of the set S follows from theorem(1) by using the contrary of this theorem.Assume that S is not minimum dominating set then ,such that is dominating set of .Therefore .Also note that for any vertex the end points of this graph of the form ( ) that adjacent to v not dominating with any vertex in (i.e. ).So clearly that is not dominating set (by the definition of dominating set).Chosen in this way ensures minimum is minimum So .

On domination number of web graph :
Theorem(3): the domination number of web graph is: Proof: For let the vertices of this graph labeled by: .(See Fig. 5) First we given the dominating set of this graph by a set isomorphic to S Where the proof of the minimality is similarly in theorem(2) So we must show that there is no proper subset is dominating set of If is any vertex in we have always two vertices of the form and not dominating with any vertex in so is not dominating set(by definition of dominating set), this ensures is minimum. .

On domination numbers of Levi graph :
As in definition(3) Levi graph can be written as where and shown in Fig. 3.So we first determine the domination number of and then we determine the domination number of Levi graph.

Lemma(1):
Proof: let the vertices of this graph labeled by: .(See Fig. 6) We given the dominating set of this graph by a set isomorphic to S Where .the proof of the minimality is as in theorem(2) So we must show that there is no proper subset dominating set of .For any , is one of the following: .We divided the set S into two sets: the vertices that is the outer circle and the inner vertices in So if we have 4 cases: Case (1): when we have is not dominating set because there is 3 vertices not dominating with any vertex in In other words there is 3 vertices in is not adjacent to any vertex in .Case (2): when we have .Also we have 2 vertices in is not adjacent to any vertex in is not dominating set.Case (3): when we have .Also we have a vertex in is not adjacent to any vertex in is not dominating set.Case (4): when we have the same case in case (2).There is 2 vertices in is not adjacent to any vertex in .Now, similarly if we have at least one vertex of the form (mod 10) not adjacent to any vertex in .So chosen S in this way ensures that there is no proper subset of S dominating G 1 minimum .

Lemma(2):
Proof: let the vertices of this graph labeled by: .(See Fig. 7) We given the dominating set of this graph by a set isomorphic to S .Where the inner vertices It is easy to see that S minimum since if S is not minimum then there is proper subset is dominating set.if , so for every vertex v the vertices which is refer to the end points of this graph not adjacent to any vertex in . is not dominating set is minimum .

Theorem(4):
Proof: The union of the graph in lemma (1) and in lemma (2) know as Levi graph; has (30) vertices labeled as same in lemma(1) and lemma (2).(See Fig. 8) We given the dominating set of this graph by a set isomorphic to S .Where .Also the proof of the minimality is as in theorem(2), assume is dominating set of Levi graph.We divided the set S into two sets: the vertices that is in the outer circle and .

If
we have the vertices in the form not dominating with any vertex in .Similarly If we have the vertices in the form not dominating with any vertex in .
is not dominating set is minimum .

Determination and testing the domination numbers of Helm graph and web graph using matlab:
In this paper we writing a programs in matlab for the purpose of making sure and certainty of the validity of our theoretical results for compute the dominating sets S and the domination number of Helm graph ,Web graph and the union of two special graphs G 1 and G 2 .Moreover we draw this results in the same graph by referring to the vertices that domination this graphs with different color from the other vertices (which colored black).
We compute examples for special cases of Helm graph and the Web graph then generalization the results for any (n) of Helm graph and for any (m) for Web graph because in every execution of this programs request an input n or m for this two graphs.

Definition( 1 )
: The Helm graph is a simple graph obtain from nwheel graph by adjoining a pendant edge at each vertices of the cycle [ ........ ... Fig.1: Helm graph We first define a prism graph then the Web graph Definition(2): A prism graph is a simple graph given by the Cartesian product graph where is a cycle with m vertices and is a path with n vertices.A web graph is a prism graph with the edges of the outer cycle removed and denoted by [10 ].

Theorem ( 1 )
:([6], p.546)A dominating set D of a graph G is minimal if and only if for each vertex one of the following conditions satisfied (i) There exist a vertex such that (ii) v is an isolated vertex in D .