ABSTRACT: Let G = (V, E) be a simple connected graph of order p and size q. If {G1, G2, …..,Gn} are edge disjoint subgraphs of G such that E(G) = E(G1) ∪ E(G2) ∪ E(G3) ∪………...∪E(Gn) then {G1, G2,…....,Gn} is said to be a Decomposition of a graph G. A graph G is said to have Double Triangular Decomposition (DTD) if G can be decomposed into two copies of {G1, G2,……..,Gn} such that each subgraph Gi is connected and |E(G_i)| = (i(i+1))/2 for 1 ≤i≤n. In this paper we investigate Double Triangular Decomposition of graphs.