Planarity Testing of Graphs on Base of a Spring Model

It is well known that planar embeddings of 3-connected graphs are uniquely determined up to isomorphy of the induced complex of nodes, edges and faces of the plane or the 2-sphere [1]. Moreover, each of the isomorphy classes of these embeddings contains a representative that has a convex polygon as outer border and has all edges embedded as straight lines. We fixate the outer polygon of such embeddings and regard each remaining edge e as a spring, its resilience being |e|k (|e| euclidean length of e, k∈ IR, 1<k<∞). For 3-connected graphs, exactly one power-balanced embedding for each k exists, and this embedding is planar if and only if the graph with the fixated border polygon has a planar embedding inside that very polygon. For k=1 or k=∞, some faces may be collapsed; we call such embeddings quasi-planar [2]. It is possible to decide the planarity of any graph embedding in linear time [3]. The motivation for this result was to develop a planarity test that simultaneously with the decision process constructs a concrete planar embedding. This algorithm should work in three steps: