String decompositions of graphs

For a graph G,if F is a nonempty subset of the edge set E (G), then the subgraph of G whose vertex set is the set of end of edges in F is denoted by [F](G). Let E(G)= boolean OR i is an element of I E, be a partition of E(G), let D-i = [E-i](G) for each i, and let phi = (D-i \ i is an element of I), then phi is called a partition of G and E-i (or D-i) is an element of phi. Given a partition phi = (D-i \i is an element of I) of G, phi is an admissible partition of G if for any vertex nu is an element of V-2 (G) there is an unique element D-i which contains vertex nu as an inner point. For two distinct vertices uand nu, u-v walk of G is a finite, alternating sequence u =u(0),e(1),u(1),e(2),...,v(n-1),e(n),u(n) = v of vertices and edges, beginning with vertex u and ending with vertex v, such that e(i) = u(i-1), u, for i = 1,2,...,n. A u-v string is a u-v walk such that no vertex is repeated except possibly u and v, i.e. u and v are allowed to appear at most two times. Given an admissible partition phi, phi is a string decomposition or SD of G if every element of phi is a string. In this paper, we prove that 2-connected graph G has an SD if and only if G is not a cycle. We also give a characterization of the graphs with cut vertices such that each graph has an SD.