1. Introduction

Domination in graphs has become an important area of research in graph theory,as evidenced by the many results contained in the two books by Haynes,Hedetniemi and Slater(1998)[6].Vernold Vivin J(2010) have studied the harmonious coloring of line graph, middle graph, central graphs of certain special graphs[12].Venketakrishnan and Swaminathan (2010)[15] have studied the colorclass domination number of middle graph and central graphs of K_1,n, C_n and P_n. In this paper we discuss (1,2)-domination in the middle and central graphs of K_1,n, C_n and P_n.

By a graph we mean a finite ,undirected graph without loops or multiple edges. A subset of is a dominating set of if every vertex of is adjacent to a vertex of . The domination number of , denoted by (), is the minimum cardinality of a dominating set of .

A (1,2) - dominating set in a graph is a set S having the property that for every vertex in there is atleast one vertex in at distance from and a second vertex in at distance atmost 2 from . The order of the smallest (1,2) - dominating set of is called the (1,2) - domination number of denoted by ^.

For a given graph of order n,the central graph is obtained, by subdividing each edge in E exactly once and joining all the nonadjacent vertices of . The central graph of a graph is an example of a split graph, where a split grah is a graph whose vertex set V can be partitioned into two sets, V1 and V2, where every pair of vertices in V1 are adjacent , and no two vertices in V2 are adjacent.

The middle graph of a graph , is the graph whose vertex set is where two vertices are adjacent if and only if they are either adjacent edges of G or one is a vertex and the other is an edge incident with it. That is, two vertices x and y in the vertex set of are adjacent in if are in and are adjacent in or is in ,y is in , and x is incident to y in . The related ideas regarding these graphs can be seen in[3,12,13,14].

2. (1,2) - domination in Middle Graphs of K_1,n, C_n and P_n.

Theorem 2.1

For any star graph

Proof:

Let and .By the definition of middle graph, we have in which the vertices induces a clique of order . Hence. Butis an independent set and each is adjacent to .

Therefore

Hence,

Theorem 2.2

For any star graph (1,2)- domination number

Proof:

Let and .By the definition of middle graph, we have in which the vertices induces a clique of order the vertex is adjacent to and is an independent set and eachSo will form a (1,2)- dominating set. That is, .

Since induces a clique,

Therefore

Theorem 2.3

For any cycle

Proof:

Let and where By the definition of middle graph , has the vertex set in which each is adjacent with and is adjacent with . In induces a cycle of length

But we know that for (Theorem 2.1,[5 ]).

Thus it is clear that ,

Theorem 2.4

For any cycle

Proof:

Let and where By the definition of middle graph , has the vertex set is adjacent with and is adjacent with . In induces a cycle of length

We have by theorem 3.1 in[ 10 ], for any cycle Cn, Hence

Theorem 2.5

For any path

Proof:

Let be a path of length and let . By the definition of middle graph, has the vertex set in which eachis adjacent to .Also

Case 1:

Then. The vertices induces a path of length

Case 2:

Then. The vertices induces a path of length

In both the cases, is a path of length That is,

But we have for , (Theorem 2.1,[5 ]). Therefore,

Theorem 2.6

For any path

Proof:

Let be a path of length and let .

By the definition of middle graph has the vertex set in which each is adjacent toand

is adjacent to .Also is adjacent to

Case 1:

Then . The vertices induces a path of length

Case 2:

Then. The vertices induces a path of length

In both the cases, is a path of length That is,

Then by theorem 2.1 in[10] we have

3. (1,2) - domination in Central Graph of K_1,n, C_n and P_n.

3.1. (1,2)-domination in C(K_1,n)

Theorem 3.1

For any star graph

Proof:

Let where deg v = n. By the definition of central graph of we denote the vertices of subdivision by That is, vv_i is subdivided by . Let and . Therefore . By the definition of central graph the subgraph induced by the vertex set and let e_ij be the edge of , connecting the vertex v_i and v_j

Since in the central graph of a star, the central vertex v together with any one of v_i’s form a dominating set. Therefore we have

Theorem 3.2

For any star graph

Proof:

Let where deg v = n. By the definition of central graph of we denote the vertices of subdivision by is subdivided by . Let and . Therefore . By the definition of central graph the subgraph induced by the vertex set is Kn and let eij be the edge of , connecting the vertex vi and vj Then . Since in the central graph of a star, the central vertex of the star is adjacent to every vertex in the central graph the vertex v together with any one of the vetices will form a {(1,2)- dominating set. Hence

3.2. (1,2)-domination in C(C_n): Consider the Following Examples

{v₁,u₂} is a dominating set and also (1,2)-dominating set.

{v₁,u₂,u₃} is a (1,2)-dominating set.

{v₁,u₂,u_3,u₄} is a (1,2)-dominating set.

{v₁,u₂,u_3,u₄,u₅} is a (1,2)-dominating set.

Theorem 3.3

For any cycle

Proof:

Let C_n be any cycle of length n and let and

By the definition of central graph has the vertex set where u_i is a vertex of subdivision of the edge v_iv_i+1 (1 ≤ i ≤ n-1) and u_n is a vertex of subdivision of the edge v_nv_1. In we can note that the vertex v_i is adjacent with all vertices except the vertices v_i+1 and v_i-1 for 1 ≤ i ≤ n-1.v₁ is adjacent with all vertices except v_n-1 and v₁.The total number of edges incident with v_i is (n-1) for every

i = 1,2,…,n and { u_i, (1 ≤ i ≤ n)} is an independent set.So { v_i}

will be a dominating set.So the dominating set will consist of (n-1) vertices.

Hence

Theorem 3.4

For any cycle

Proof:

Let C_n be any cycle of length n and let and .

By the definition of central graph has the vertex set where u_i is a vertex of subdivision of the edge v_iv_i+1 (1 ≤ i ≤ n-1) and u_n is a vertex of subdivision of the edge v_nv_1. In we can note that the vertex v_i is adjacent with all vertices except the vertices v_i+1 and v_i-1 for 1 ≤ i ≤ n-1.v₁ is adjacent with all vertices except v_n-1 and v₁.The total number of edges incident with v_i is (n-1) for every

i = 1,2,…,n and { u_i, (1 ≤ i ≤ n)} is an independent set. So { v_i}

will be a dominating set. But every minimum cardinality dominating set is also a (1,2)-dominating set in

Hence

3.3. (1,2)-domination in C(P_n)

v₁,u₂} is a dominating set and also (1,2)-dominating set.

.

{v₁,u₂,u₃} is a (1,2)-dominating set.

{v₁,u₂,u_3,u₄} is a (1,2)-dominating set.

Theorem 3.5

For any path P_n γ (C(P_n)) = n-1

Proof.

Let P_n be any path of length n-1 with vertices v₁,v₂,...........,v_n. On the process of centralisation of P_n, let u_i be the vertex of subdivision of the edges v_iv_i-1 (1 ≤ i ≤ n).

Also let v_iu_i=e_i and u_iv_i+1= (1 ≤ i ≤ n-1).

By the definition of central graph the non-adjacent vertices v_i and v_j of P_n are adjacent inC(P_n) by the edge e_ij. Therefore V(C(P_n)) = {v_i/1 ≤ i ≤ n} ∪ {u_i/1 ≤ i ≤ n-1} and E(C(P_n))={e_i :1≤ i ≤ n-1}∪ {:1≤i≤n-1} ∪ {e_ij : 1 ≤ i ≤ n -2, i+2 ≤ j ≤ n}. In C(P_n) we can see that the vertex v_i is adjacent with all vertices except the vertices v_i+1 and v_i-1 for 1 ≤ i ≤ n-1.v₁ is adjacent with all vertices except v_2. { u_i, (1 ≤ i ≤ n)} is an independent set So So { v_i}

will be a dominating set.

γ(C(P_n))= n-1.

Theorem 3.6

For any path P_n,γ_(1,2) ((C(P_n))= n-1.

Proof:

Let P_n be any path of length n-1 with vertices v₁,v₂,...........,v_n. On the process of centralisation of P_n, let u_i be the vertex of subdivision of the edges v_iv_i-1 (1 ≤ i ≤ n).

Also let v_iu_i=e_i and u_iv_i+1= (1 ≤ i ≤ n-1).

By the definition of central graph the non-adjacent vertices v_i and v_j of P_nare adjacent inC(P_n) by the edge e_ij. Therefore V(C(P_n)) = {v_i/1 ≤ i ≤ n} ∪ {u_i/1 ≤ i ≤ n-1} and E(C(P_n))={e_i :1≤ i ≤ n-1}∪ {:1≤i≤n-1} ∪ {e_ij : 1 ≤ i ≤ n -2, i+2 ≤ j ≤ n}. In C(P_n) we can see that the vertex v_i is adjacent with all vertices except the vertices v_i+1 and v_i-1 for 1 ≤ i ≤ n-1.v₁ is adjacent with all vertices except v_2. { u_i, (1 ≤ i ≤ n)} is an independent set SoSo { v_i}

will be a dominating set.

γ(C(P_n))= n-1.

But every minimum cardinality dominating set of the central graph of a path is also a (1,2)- dominating set. Hence γ_(1,2) (C(P_n)) = n-1.

4. Relation between Domination Number and (1,2)-domination Number in Middle and Central Graph of Stars, Cycles and Paths

Theorem 4.1

In the middle graph of a star, ,the domination number equals the (1,2)-domination number.

Proof:

This result is obvious from theorem 2.1 and 2.2.

Theorem 4.2

In the middle graph of cycles, the domination number equals the (1,2)-domination number.

Proof:

This result is obtained by theorem 2.3 and theorem 2.4.

Theorem 4.3

In the middle graph of paths, , the domination number is less than or equal to the (1,2)- domination number.

Proof:

This result is obtained by theorem 2.5 and theorem 2.6.

Theorem 4.4

In the central graph of any star, C(K_1,n), the domination number equalsthe (1,2)-domination number.

Proof:

This is clear from theorem 3.1 and 3.2.

Theorem 4.5

In the central graph of cycles, C(C_n), the domination number equals the (1,2)- domination number.

Proof:

This result is due to the theorem 3.3 and 3.4.

Theorem 4.6

In the central graph of paths,C(P_n), the domination number equals the (1,2)- domination number.

Proof:

This result is obtained from theorem 3.5 and 3.6.

5. Conclusions

In this paper we have extended (1,2)- domination to the middle graph and central graph of stars, cycles, and paths and discussed both domination and (1,2)- domination number of these graphs. In all cases it is important to see that the domination number is less than or equal to the (1,2)- domination which coincides the result established in[8].