TOTAL CO-INDEPENDENT DOMINATION OF JUMP GRAPH

A set D ⊆ V [J(G)] is dominating set of jump graph, if every vertex not in D is adjacent to vertex in D. The domination number of the jump graph is the minimum cardinality of dominating set of jump graph J(G). A total dominating set S of a graph G is called Co-independent dominating set if the induced subgraph 〈G〉 has no edge and has at least one vertex. In this paper, we introduce the total Co-independent domination of jump graph, and its exact values for some standard graphs and some results are established. AMS Subject Classification: 05C78


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, we use X to denote the subgraph induced by the set of vertices X.A vertex of degree one is called a 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).The circumference of a graph G with at least one cycle is the length of a longest cycle in G and it is denoted by circum(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.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.For terminology and notations not specifically defined here we refer reader to [1] [2].For more details about domination number and its related parameters, we refer to [3][4] [5].
A dominating set S of G is called a connect dominating set if the induced subgraph S is connected.The minimum cardinality of a connected dominating set of G is called the connected domination number of G and is denoted by γ c (G) [6].A dominating set S of G is called total dominating set if the induced subgraph S has no isolated vertices.The minimum cardinality of a total dominating set of G is called the total domination number of G and is denoted by γ t (G) [3].
Definition 1.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 [7].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.
Remark 2. The isolated vertices of G (if G has) play no role in line graph and jump graph transformation.Here we assume that the graph G under consideration is non-empty and has no isolated vertices found in [7].
{2, 3} is the minimum total dominating set.Hence γ t (G) =2, clearly {2, 3} is not total Co-independent dominating set of G.The minimum total Coindependent dominating sets are {1, 2, 3, 4}, {1, 2, 3, 5}, {2, 3, 4, 9}, {2, 3, 5, 9}.Therefore γ t,coi (G) = 4.As in the standard dominating set, any minimum total Co-independent dominating set is minimal, but the converse is not true as seen in figure 1.The set {1, 2, 4, 5, 6, 7} is minimal total Co-independent dominating set but not minimum total Co-independent dominating set.Definition 3. A total dominating set S of a J(G) is called total Co-Figure 1 independent dominating set if the induced subgraph V − S has no edge and has at least one vertex.The minimum cardinality of a total Co-independent dominating set of J(G) is called the total Co-independent domination number of J(G) and is denoted by γ t,coi J(G) .

Main Results
Observation.For any Cycle C p , with p ≥ 5, γ t,coi [J(C p )] = 4. Observation.For any path P p , with p ≥ 5, γ t,coi [J(P p )] = 4. Observation.For any complete graph K p , with p ≥ 5, γ t,coi [J(K p )] = 6.Observation.For any connected (p, q) graph G, γ t,coi [J(G)] = q.Observation.For any complete bipartite graph K m,n 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.
Hence V 1 is a minimum total Co-independent dominating set.So the total Co-independent domination number of the jump graph will be equal to the degree of V 1 we say γ t,coi [J(G)] > 4. Case (ii): 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 , v 2 ⊆ V (J(G)).These two vertices form total Co-independent dominating set in jump graph.Since these vertices v 1 , v 2 are adjacent to all other vertices of V (J(G)),it follows that v 1 , v 2 becomes a minimum total Co-independent dominating set.Hence γ t,coi [J(G)] = 4.In view of above cases, we can conclude that for any connected graph G, γ t,coi [J(G)] ≥ 4.

Theorem 5. For any connected graph G with circumference
We can prove the theorem with the following cases.Case i: For Circum(G) = 4,choose a vertex V 1 that have longest cycle among others.Let V 1 = {v 1 , v 2 , . . .} corresponding to the elements of {e 1 , e 2 , . . .} forming a total Co-independent dominating set in jump graph J(G).Every vertex u / ∈ V 1 is adjacent to a vertex in V 1 .Hence V 1 is a minimum total Coindependent dominating set.So the total Co-independent domination number of the jump graph will be equal to the longest cycle.We say γ t,coi [J(G)] > 4.
Case ii: For Circum(G) > 4.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 , v 2 } ⊆ V (J(G)).These two vertices form total Co-independent dominating set in jump graph.Since these vertices {v 1 , v 2 } are adjacent to all other vertices of V (J(G)), it follows that {v 1 , v 2 } becomes a minimum total Co-independent dominating set.Hence γ t,coi [J(G)] = 4.In view of above cases, we can conclude that for any connected graph G, γ t,coi [J(G)] ≥ 4. Theorem 6.For any tree T with diameter greater than 3, γ t,coi [J(T )] = q.
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 i , v k } of J(T ) corresponding to the edges of {e i , e k } in T will form total Co-independent dominating set in

Conclusion
Thus we conclude the total Co-independent domination number and the parameters of total Co-independent domination number of jump graphs like k p , C p ,P p , W p and K m,n .