Necessary and sufficient condition for a graph to be three-terminal series-parallel

A "three-terminal series-parallel graph" is defined to be a three-terminal graph which is constructed by means of repeating only specified series and parallel connections. This definition is based upon our previous work about transformerless interconnections of two-port networks. Some properties of the graph are shown under the foregoing definition and, especially, a theorem is given stating that a three-terminal graph is three-terminal series-parallel if and only if neither of two certain graphs can be obtained from it by opening or shorting some edges.