Domination Number of Jump Graph

A set D ⊆ V [J(G)] is dominating set of jump graph, if every vertex not in D is adjacent to a vertex in D. The domination number of the jump graph is the minimum cardinality of dominating set of jump graph J(G). We also study the graph theoretic properties of γ[J(G)] and its exact values for some standard graphs.The relation between γ[J(G)] with other parameters are also investigated. Mathematics Subject Classification: 05C69, 05C70, 05C76


Introduction
Let G(p, q) be a graph with p =| V | and q =| E | denote the number of vertices and edges of a graph G, respectively.All the graphs considered here are finite, non-trivial, undirected and connected without loops or multiple edges.
In general, the degree of a vertex v in a graph G is the number of edges of G incident with v and it is denoted by degv.The maximum (minimum) degree among the vertices of G is denoted by Δ(G) (δ(G)).We denote the minimum number of edges in edge cover of G ( i.e., edge cover number ) by α 1 (G) and the minimum number of edges in independent set of edges of G (i.e., edge independent set) by β 1 (G).The subgraph induced by X ⊆ V is denoted by X .A vertex of degree one is called an pendent vertex.A vertex adjacent to pendent vertex is called the support vertex.The maximum d(u, v) for all u in G is eccentricity of v and the maximum eccentricity is the diameter diam(G).As usual, P n , C n , and K n , are respectively, the path, cycle and complete graph of order n.K r,s is the complete bipartite graph with two partite sets containing r and s vertices.Any undefined term or notation in this paper can be found in Harary [2].

Preliminary Notes
The line graph L(G) of G has the edges of G as its vertices which are adjacent in L(G) if and only if the corresponding edges are adjacent in G.We call the complement of line graph L(G) as the jump graph J(G) of G, found in [1].The jump graph J(G) of a graph G is the graph defined on E(G) and in which two vertices are adjacent if and only if they are not adjacent in G. Since both L(G) and J(G) are defined on the edge set of a graph G, it follows that isolated vertices of G (if G has ) play no role in line graph and jump graph transformation.We assume that the graph G under consideration is nonempty and has no isolated vertices [1] .

Definition 2.1 We now define the domination number of jump graph. Let G = (V, E) be a graph. A set D ⊆ V is said to be a dominating set, if every vertex not in D is adjacent to a vertex in D . The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set. Analogously, a set D ⊆ V [J(G)] is said to be dominating set of J(G) , if every vertex not in D is adjacent to a vertex in D. The domination number of Jump graph , denoted by γ[J(G)], is the minimum cardinality of a dominating set in J(G).
For any graph G, with p ≤ 4, the jump graph J(G) of G, is disconnected.Since we study only the connected jump graph, we choose p > 4 [3] .We recall following classical theorems to prove our results.Theorem 2.2 [4] : Let G be a graph without isolated vertices.If D is a minimal dominating set, then V − D is a dominating set.

Main Results
We state some preliminary result in the following theorem for some standard graphs.

For any Complete bipartite graph
for K m,n where m,n≥ 3.

For any Wheel
Proof of the theorem is obvious.
Proof.Since every vertex in V [J(G)] − D is adjacent to at least one vertex in D, there will be a contribution from each vertex of V [J(G)] − D by one to the sum of degrees of vertices of D. Hence the proof of the theorem.
. ., v n } be the set of vertices in J(G) corresponding to the set of independent edges {e 1 , e 2 , e 3 , . . ., e n } of G .By the definition of J(G), the elements of The following theorem gives the relationship between domination number of a graph with its jump domination number of a graph. 2 .But for any simple graph G, q ≤ p(p−1)

.
From the above equations, we get

Proof. Let uv be a path of maximum distance in G. Then d(u, v) = diam(G).
We can prove the theorem with the following cases.Case 1.For diam(G) = 2, choose a vertex v 1 of eccentricity 2 with maximum degree among others.Let V 1 = {v 1  1 , v 1 2 . . .} corresponding to the elements of {e 1 , e 2 . . .} forming a dominating set in Jump graph J(G).
a minimum dominating set.So domination number of the jump graph will be equal to the degree of v 1 , also by theorem (3.2), we say γ[J(G)] > 2.
Case 2. For diam(G) > 2 , let e 1 be any edge adjacent to u and e 2 be any edge adjacent to v. Let {e 1 , e 2 } ⊆ E(G) form a corresponding vertex set {v 1  1 , v 1 2 } ⊆ V (J(G)).These two vertices form a dominating set in jump graph.Since these vertices {v 1  1 , v 1 2 } are adjacent to all other vertices of V (J(G)), it follows that {v 1 , v 2 } becomes a minimum dominating set.Hence γ[J(G)] = 2.In view of above cases, we can conclude that for any connected graph Similar type of result can be proved for any tree T.

Theorem 3.7 For any tree T with diameter greater than 3 ,γ[J(T )] = 2.
Proof.If the diameter is less than or equal to 3, then the jump graph will be disconnected.Let uv be a path of maximum length in a tree T where diameter is greater than 3 .Let e i be the pendant edge adjacent to u and e k be the pendant edge adjacent to v. The vertex set {v 1  i , v 1 k } of J(T ) corresponding to the edges of {e i , e k } in T will form the dominating set in J(T ).Since all the other vertices of V [J(T )] are adjacent with {v 1 i , v 1 k }, it forms a minimum dominating set.Hence γ[J(T )] = 2. Theorem 3.8 For any connected (p, q) graph G, γ[J(G)] ≤ q −Δ(G) where Δ(G) is the maximum degree of G.
Proof.An edge {e i } in any connected graph G is adjacent to at least one more edge in G.In Jump graph, the vertex {v 1 i } corresponding to {e i } is non adjacent to {v k i , v j i } of e k , e i in J(G).Therefore by definition of domination number of graph γ(G), the dominating set contains at least two elements.Hence γ[J(G)] ≥ 2.
(1) Let E be the set of edges in G. Then E = V [J(G)].Suppose D= {v 1 , v 2 , v 3 , . . ., v k } be the dominating set.Then V − D is also a dominating set.One among these two sets will form a minimal dominating set.So by the definition of domination number of graph, we can say domination number, γ[J(G)] of jump graph is given by From ( 1) and ( 2), we get 2 ≤ γ[J(G)] ≤ q 2 .

Theorem 3.10 For any connected graph G without pendent vertex, γ[J(G)] ≤ δ(G).
Proof.Let V= {v 1 , v 2 , v 3 , . . ., v n } be the set of vertices in G and v 1 be one among the vertices with minimum degree.Let {e 1 , e 2 , e 3 , . . ., e k } be the set of edges adjacent to v 1 in G. Then E 1 ⊆ V [J(G)] will form the dominating set in J(G) .So | E 1 |= δ(G).Obviously it becomes the minimum dominating set.Therefore γ[J(G)] ≤ δ(G).

2 2 . 2 ,
Proof.For any (p, q) graph G, we have γ(G) ≤ min{| D |, | V − D |}, ≤ p by virtue of theorem B. Further, by the definition of Jump graph, we have V (J(G)) = q.Hence, γ[J(G)] ≤ q one of the vertices with maximum degree.By definition of jump graph, E(G) = V [J(G)].Consider I = {e 1 , e 2 , e 3 , . . .e k } as the set of edges adjacent to v 1 in G. Let H ⊆ V (J(G)) be the set of vertices of J(G).such that, H ⊆ E − I. Then H itself form a minimally dominating set.Therefore γ