Planar, coplanar, and totally planar n-port networks

The aim of this paper is to provide necessary and sufficient conditions for an -port network to be planar, coplanar, and totally planar. Generally, a planar graph does not have a unique plane representation, and a combinatorial problem arises in determining whether the graph is planar, coplanar, or totally planar. To avoid the combinatorial complexity the graph is decomposed into 3-connected components called atoms. Since atoms have unique sets of boundaries and coboundaries, our planarity and coplanarity conditions, which are established on atoms, may be easily examined on any plane representations of the atoms.