Series - parallel graphs and depth-first search trees

Series-parallel graphs play a significant role in the analysis and synthesis of electrical networks, communication networks, and switching circuits. On the other hand, due to their very tractable structure, a number of algorithmic problems which are NP-complete for arbitrary graphs can be efficiently solved for the special case of series-parallel graphs, see some recent results in [7]. In another recent work, Shinoda et al. [6] have shown that series-parallel graphs can be completely characterized by a property of their spanning trees. They proved, that every spanning tree of a series-parallel graph is a depth-first search (or DFS) tree of a 2-isomorphic copy of . The proof in [6] is, however, existential. The purpose of this note is to provide a constructive proof of this property. We present a procedure which for a given series-parallel graph and its spanning tree , produces a 2-isomorphic copy of such that the edges of generate in a DFS tree of . Our considerations are entirely based on the classical constructive definition of series-parallel graphs and the Duffin\´s characterization, unlike the work [6], where several other characterizations of these graphs are utilized. We refer the reader to Chen [1] and Harary [4] for graph-theoretic terms not defined here.