Let \({C}_{n}, {K}_{n},{W}_{n},{K}_{r,s}\) denote a cycle, complete graph, wheel graph, complete bipartite graph respectively. An edge cycle graph of a graph \(G\) is the graph \(G({C}_{k})\) formed from one copy of \(G\) and \(|E(G)|\) copies of \({P}_{k},\) where t he ends of the \({i}^{th}\) edge are identified with the ends of \({i}^{th}\) copy of \({P}_{k}\) . In this article, we determine the necessary and sufficient conditions for the existence of paw- decompositions of the diamond graph \({Br}_{n}\) and some edge cycle graphs like \({K}_{n}\left({C}_{3}\right), { W}_{n}\left({C}_{3}\right),{ K}_{r,s}\left({C}_{3}\right), { C}_{n}\circ \stackrel{\leftharpoonup}{{K}_{m}}({C}_{3})\) and \({P}_{n}\circ \stackrel{\leftharpoonup}{{K}_{m}}({C}_{3})\) where \(\circ \) denotes the corona of graphs.
