CONTINUOUS MONOTONIC DECOMPOSITION OF JUMP GRAPH OF PATHS AND COMPLETE GRAPHS

The Jump graph J(G) of a graph G is the graph whose vertices are edges of G and two vertices of J(G) are adjacent if and only if they are not adjacent in G. In this article, we have given characterization for the Jump graph of paths into Continuous monotonic star decomposition. Also we have given characterization for the Jump graph of complete graphs into Continuous monotonic tree decomposition.


INTRODUCTION
Let G = (V, E) be a simple undirected graph without loops or multiple edges. A path on n vertices is denoted by P n , cycle on n vertices is denoted by C n and complete graph on n vertices is denoted by K n . The neighbourhood of a vertex v in G is the set N(v) consisting of all vertices that are adjacent to v. |N(v)| is called the degree of v and is denoted by d(v). A complete bipartite graph with partite sets V 1 and V 2 , where |V 1 | = r and |V 2 | = s, is denoted by K r,s . The graph graph H of G, then the decomposition is called a H-decomposition of G. A decomposition, {G 1 , G 2 , . . . , G k } for all k ∈ N is said to be a Continuous Monotonic Decomposition (CMD) if each G i is connected and |E(G i )| = i for all i ∈ N. The concept of CMD was introduced by Paulraj Joseph and Gnanadhas [4].
The Jump graph J(G) of a graph G is the graph whose vertices are edges of G and two vertices of J(G) are adjacent if and only if they are not adjacent in G. Equivalently complement of line graph L(G) is the Jump graph J(G) of G. This concept was introduced by Chartrand in [1]. Coconut tree CT (m, n) is a graph obtained from the path P n by appending m new pendant edges at an end vertex of P n . Double coconut tree D(n, r, m) is a graph obtained by attaching n > 1 pendant vertices to one end of the path P r and m > 1 pendant vertices to the other end of the path P r .

CONTINUOUS MONOTONIC STAR DECOMPOSITION OF JUMP GRAPH OF PATHS
Let J(P n ) denote the Jump graph of paths. Then J(P n ) is a connected graph if and only if n ≥ 5. Let us consider the connected jump graph of paths. Let the edges of path P n be labelled as x 1 , x 2 , . . . , x n−1 . Since the number of edges of path P n is (n − 1), the number of vertices of J(P n ) is (n − 1). The number of edges of Jump graph of paths J(P n ) is n−2 2 .
Proof. Let m ≥ 2. and n = m + 3. Let us prove this lemma by induction on m.
Assume that the result is true for m − 1. Clearly  Thus m = n − 3. Hence (i).
Clearly {S 1 , S 2 , . . . , S 9 } forms a CMSD of J(P 12 ). Since the number of edges of complete graphs K n is n(n−1) 2 , the number of vertices of J(K n ) is Define

CONTINUOUS MONOTONIC TREE DECOMPOSITION
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Thus we get T 4 copies of S 3 . Proceed like this.