Independent Transversal Equitable Domination in Graphs

A set S ⊆ V of vertices in a graph G = (V,E) is called an equitable dominating set if for every vertex u in V − S there exists at least one vertex v in S adjacent to u and |deg(u) − deg(v)| ≤ 1. An equitable dominating set which intersects every maximum independent set in G is called an independent transversal equitable dominating set. The minimum cardinality of an independent transversal equitable dominating set is called the independent transversal equitable domination number of G and is denoted by γite(G). In this paper we begin an investigation of this parameter. Mathematics Subject Classification: 05C69


Introduction
By a graph G = (V, E) we mean a finite and undirected graph with no loops and multiple edges.As usual p = |V | and q = |E| denote the number of vertices and edges of a graph G, respectively.In general, we use X to denote the subgraph induced by the set of vertices X. N(v) and N [v] denote the open and closed neighbourhood of a vertex v, respectively.A set D of vertices in a graph G is a dominating set if every vertex in V − D is adjacent to some vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G.A line graph L(G) (also called an interchange graph or edge graph) of a simple graph G is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge if and only if the corresponding edges of G have a vertex in common.A matching M in G is a set of pairwise nonadjacent edges; that is, no two edges share a common vertex.The matching number is the maximum cardinality of a matching of G and is denoted by β 1 (G).The independent transversal domination was introduced by I. S. Hamid [3].A dominating set D in a graph G which intersects every maximum independent set in G is called independent transversal dominating set of G.The minimum cardinality of an independent transversal dominating set is called independent transversal domination number of G and is denoted by γ it (G).For terminology and notations not specifically defined here we refer reader to [4] For more details about equitable domination number see [11]. .

Independent Transversal equitable Domination Number
In this section, we determine the value of independent transversal equitable domination number for some standard families of graphs such as paths, cycles, complete bipartite and wheels.Also we determine γ ite (G) for disconnected graphs.Obviously any minimum independent transversal equitable dominating set D is also minimal, but the converse is not true as illustrated in the next example.
Example 2.3.Let G be a graph as in Figure 1.
There is only one maximum independent sets which is {v If G = (V, E) be any graph, then V is independent transversal dominating set of G. Thus, the independent transversal domination number is defined for any graph G The following inequality shows the relation between the domination number and degree equitable domination number an the independent transversal equitable domination number of a graph G.
Proof.Let G be a graph with minimum independent transversal equitable dominating set D, then D is an equitable dominating set of G and any equitable dominating set is also dominating set of G and from the definitions of the parameters γ(G), γ e (G) and Proof.Let G be a graph with minimum independent transversal equitable dominating set S. From the definition of the independent transversal dominating set of G, S is also independent transversal dominating set, and any independent transversal dominating set is dominating set of Theorem 2.6.[11] If G is regular or (k; k + 1) bi-regular graph , for some k, then γ e (G) = γ(G).
Proof.Let G be any regular graph and let D be an independent transversal dominating set of G, such that |D| = γ it (G).Then D is dominating set which intersect every maximum independent set of G and by theorem 2.6 D is equitable dominating set intersect every maximum independent set of G.
Similarly by using Theorem 2.6 and Proposition 2.5 and by the same way we can prove that if G is (k; k + 1) bi-regular graph , for some k ≥ 0, then We generalize the Theorem 2.6 by the following result.

Theorem 2.8. For any graph G which all of its edges are equitable, γ it (G) = γ ite (G).
Proof.Let G be a graph G = (V, E) such that for any edge e ∈ E e is equitable edge.Then any dominating set of G will also be equitable dominating set of G. Suppose that D is an independent transversal dominating set of G with size |D| = γ it (G) .Then D is also an independent transversal equitable dominating set, that means γ it (G) = γ ite (G) and by Proposition 2.5, we have The converse of Theorem 2.8 is not true in general as illustrated in the following example.
Example 2.9.Let G be a graph as in Figure 2, we have but not all the edges are equitable edges Theorem 2.10.[3] For any path P p with p ≥ 3, we have Proof.Let G = (V, E) be a graph with at least one isolated vertex say v and S is a minimum equitable dominating set of G that is γ e (G) = |S| .Obviously the vertex v must be belong to any equitable dominating set and also to any maximum independent set of G. Therefore the minimum equitable dominating set S is is intersect every maximum independent set of G. Hence S is minimum independent transversal equitable dominating set of G. Thus,γ ite (G) = γ e (G).Proof.If G = (V, E) be any graph with p vertices, then V (G) is an independent transversal equitable dominating set of G. Therefore γ ite (G) ≤ p and also if the graph contains only one vertex we have There are many graphs of order p other than p 3 , see the following example.
Example 2.17.Let G be graph as in Figure 3.
There is only two equitable dominating sets of G, There are two maximum independent sets Proof.Let H be a complete graph of order a ≥ 3. Let G be the graph obtained from H by attaching pendent edge in each vertex of H. Let V (H) = {v 1 , v 2 , ..., v a } and let the pendent vertices are u 1 , u 2 , ..., u a .Now clearly any minimum equitable dominating set of G contains all the pendent vertices u 1 , u 2 , ...u a and one vertex from the complete graph that the minimum dominating set of G is of the form u 1 , u 2 , ...u a , v i such that i ∈ {1, 2, ..., a} that is γ e (G) = a + 1, and there is only one maximum independent set u 1 , u 2 , ...u a .Hence γ ite (G) = γ e (G) = a + 1.   Proof.Let G = (V, E) be any connected graph, with δ e (G) = 1.Then by Theorem 2.20.γ ite (G) ≤ γ e (G)+1.and from [11] if G has no equitable isolated vertices, we have γ e ≤ p 2 .Hence γ ite (G) ≤ p+2 2 .
The minimum cardinality of such a dominating set is denoted by γ e (G) and is called equitable domination number of G. D is minimal if for any vertex u ∈ D, D − {u} is not a equitable dominating set of G.A subset S of V is called a equitable independent set, if for any u ∈ S, v / ∈ N e (u), for all v ∈ S −{u}.If a vertex u ∈ V be such that |deg(u) −deg(v)| ≥ 2 for all v ∈ N(u) then u is in each equitable dominating set.Such vertices are called equitable isolates.The equitable neighbourhood of u denoted by N e (u) is defined as N e (u) = {v ∈ V /v ∈ N(u), |deg(u) − deg(v)| ≤ 1} and u ∈ I e ⇐⇒ N e (u) = ∅.The cardinality of N e (u) is denoted by deg e (u).The maximum and minimum equitable degree of a point in G are denoted respectively by Δ e (G) and δ e (G).That is Δ e (G) = max u∈V (G) |N e (u)|, δ e (G) = min u∈V (G) |N e (u)|.An edge e = uv called equitable edge if |deg(u) − deg(v)| ≤ 1.

Theorem 2 . 15 .
For any graph G = (V, E) with p vertices, 1 ≤ γ ite (G) ≤ p.Further γ ite (G) = p if and only if either G = K p or for any vertex v ∈ V (G), deg e (v) = 0

p and p ≥ 2 .
Now if the maximum equitable independence number β e (G) ≥ 2, then for some vertex v ∈ V (G) we have V (G) − {v} is independent transversal equitable dominating set, that means γ ite (G) ≤ p − 1 a contradiction.Therefore β e (G) = 1.Hence G = K p or for any vertex v ∈ V (G), deg e (v) = 0.Theorem 2.16.Let G be a connected graph with p vertices such that δe (G) ≥ 1.Then γ ite (G) = p − 1 if and only if G ∼ = P 3 .Proof.If G ∼ = P 3 , then obviously γ ite (G) = 3, that is γ ite (G) = p − 1.Conversely, suppose that G be a connected graph with p vertices such that δ e (G) ≥ 1.Then the maximum equitable independence number β e (G) ≥ 2. Now suppose that there exist two equitable adjacent vertices u and v of degree two.Then the set S = V (G) − {u, v} is an independent transversal equitable dominating set of G and hence γ ite (G) ≤ p − 2, a contradiction.Hence for any equitable adjacent vertices u and v either deg e (u) = 1 or deg e (v) = 1 and γ ite (G) = p − 1. Hence G ∼ = P 3 .

Theorem 2 .
20.For any graph G = (V, E), γ ite (G) ≤ δ e (G) + γ e (G).Proof.Let G = (V, E) be a graph and v be any vertex in V (G) such that deg e (v) = δ e (G) and let D be minimum equitable dominating set of G. Then every maximum independent set of G contain a vertex of N e [u], so D ∪ N e [u] is independent transversal equitable dominating set.We have also S intersects N e [u].Therefore |D ∪ N e [u]| ≤ δ e (G) + γ e (G).Hence γ ite (G) ≤ δ e (G) + γ e (G).
From Theorem 2.10, Theorem 2.11 and Theorem 2.7 the following results are immediate.For any path P p with p ≥ 3, we have p 3 , otherwise.p 3 , otherwise.Theorem 2.14.Let G = (V, E) be any graph with at least one isolated vertex, then γ ite (G) = γ e (G).