Planarity Testing of Graphs on Base of a Spring Model
Hotz, Günter and Lohse, Steffen (2002) Planarity Testing of Graphs on Base of a Spring Model. In: Graph Drawing 9th International Symposium, GD 2001, September 23-26, 2001, Vienna, Austria , pp. 471-472 (Official URL: http://dx.doi.org/10.1007/3-540-45848-4_51).
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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 . 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 $\vert e\vert^k$ ($\vert e\vert$ euclidean length of $e$, $k\!\in\!\bbbr, 1\!<\!k\!<\!\infty$). 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 $k\!=\!\infty$, some faces may be collapsed; we call such embeddings quasi-planar . It is possible to decide the planarity of any graph embedding in linear time . 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: 1. Choose any circle of the graph and embed it as a convex polygon in the plane or as an equator of the 2-sphere. 2. Fixate this circle and hang in the rest of the graph. Substitute each edge by a spring of a fixed norm $k$ and compute its power-balanced position. 3. Decide if the computed embedding can be decomposed into two planar flanks. The graph is planar if and only if this is possible. On the sphere, these flanks should be represented by the two hemispheres belonging to the fixated circle.
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