Lecture Notes in Computer Science Volume 2265, 2002, pp 471-472

Planarity Testing of Graphs on Base of a Spring Model

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Abstract

It is well known that planar embeddings of 3-connected graphs are uniquely determined up to isomorphyof the induced complex of nodes, edges and faces of the plane or the 2-sphere [1]. Moreover, each of the isomorphyclasses 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 anygraph embedding in linear time [3]. The motivation for this result was to develop a planaritytest that simultaneouslywith the decision process constructs a concrete planar embedding. This algorithm should work in three steps: