TWO-OUT DEGREE EQUITABLE DOMINATION IN THE MIDDLE, CENTRAL AND THE LINE GRAPHS OF PN, CN AND K1,N

Let G=(V,E) be a simple, finite, connected and undirected graph. A dominating set D of G is said to be two-out degree equitable dominating set if for any two vertices u, v ∈ D such that│odD u − odD(v)│ ≤ 2 , where odD u = │N v ∩ (V − D). The minimum cardinality of two -out degree equitable dominating set is called twoout degree equitable domination number and it is denoted by γ2oe(G). In this paper, we introduced the two-out degree equitable domination numbers in the middle, central and the line graphs of the path Pn, cycle Cn and star K1,n graphs.


INTRODUCTION
The concept of domination was first studied by Ore and Berge (1962). A non-empty set D  V is said to be a dominating set of G if every vertex in V-D is adjacent to atleast one vertex in D. The minimum cardinality of the minimal dominating set D is = called the domination number and it is denoted by γ(G).
An equitable domination has interesting application in the context of social network. In a network, nodes with nearly equal capacity may interact with each other in a better way. In society, persons with nearly equal status, tend to be friendly. Ali Sahal and V.Mathad (Sahal,2013) introduced the concept of two out degree equitable domination in graphs. In this paper, we investigated the two out degree equitable domination number in the middle and the central graphs of Pn , Cn and K1,n graphs.

Definition 1.1 (8)
The middle graph of a connected graph G denoted by M(G) is the graph whose vertex set is V(G)UE(G) where two vertices are adjacent if (i) They are adjacent edges of G (or) (ii) One is a vertex of G and the other is an edge incident with it.

Definition 1.2 (8)
For a given graph G=(V,E) of order n, the central graph C(G) is obtained, by subdividing each edge in E exactly once and joining all the non adjacent vertices of G. The central graph C(G) of a graph G is an example of a split graph, where a split graph is a graph whose vertex set V can be partitioned into two sets, V1 and V2, where each pair of vertices in V1 are adjacent, and no two vertices in V2 are adjacent.

Definition 1.3 (7)
A dominating set D in a graph G is called a two-out degree equitable dominating set if for any two vertices , ∈ such that │ − ( )│ ≤ 2, where = │ ∩ − │. The minimum cardinality of a two-out degree equitable dominating set is called the two-out degree equitable domination number of G and is denoted by γ2oe(G).
In the consequent section, we obtained the two-out degree equitable domination number γ2oe(G) of the middle graph of Pn, Cn and K1,n graphs.

Definition 1.4 (9)
Line graph L(G) of a graph G is defined with the vertex set E(G), in which two vertices are adjacent if and only if the corresponding edges are adjacent in G.

Two-out degree Equitable domination in the Middle graphs of
, and , .

Example 2.1
Let G be the middle graph as in the figure.
.By the definition of central graph, the non-adjacent vertices of are adjacent in .

Two-out degree Equitable domination in the Central graphs of
, and , .
In this section, we obtained the two-out degree equitable domination number 2 ( ) of the central graphs of the path , cycle and the star graph 1, .

Example 3.1.
Let G be the central graph as in the figure. we obtained the two-out degree equitable domination number.
The line graph of , itself.
Since the degree of any vertex in ( ) is 2 except the initial and terminal vertices.

CONCLUSION
In this paper, we introduced two-out degree equitable domination number in the middle, central and line graph of , , 1, graphs. We extend this study on some more special classes of graphs.