A necessary and sufficient condition for any tree of a connected graph to be a DFS-tree of one of its 2-isomorphic graphs

It is shown that any tree of a connected graph G is a depth-first-search (DFS)-tree of one of its 2-isomorphic graphs if, and only if, G is a series-parallel graph, where two graphs are said to be 2-isomorphic if they have the same set of edges as well as the same set of circuits. Only nonseparable graphs are considered. A basic theorem on series-parallel graphs is given, and the main theorems are discussed.<>