Journal of Graph Algorithms and Applications
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Special Issue on Selected Papers from the Seventeenth International Symposium on Graph Drawing, GD 2009
Geometric Simultaneous Embeddings of a Graph and a Matching
Sergio Cabello, Marc van Kreveld, Giuseppe Liotta, Henk Meijer, Bettina Speckmann, and Kevin Verbeek
Vol. 15, no. 1, pp. 79-96, 2011. Regular paper.
Abstract The geometric simultaneous embedding problem asks whether two planar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous embedding is a current topic in graph drawing and positive and negative results are known for various classes of graphs. So far only connected graphs have been considered. In this paper we present the first results for the setting where one of the graphs is a matching. In particular, we show that there exist a planar graph and a matching which do not admit a geometric simultaneous embedding. This strengthens an analogous negative result for a planar graph and a path. On the positive side, we describe algorithms that compute a geometric simultaneous embedding of a matching and a wheel, outerpath, or tree. Our drawing algorithms minimize the number of orientations used to draw the edges of the matching. Specifically, when embedding a matching and a tree, we can draw all matching edges horizontally. When embedding a matching and a wheel or an outerpath, we use only two orientations.
Submitted: November 2009.
Reviewed: August 2010.
Revised: September 2010.
Accepted: October 2010.
Final: November 2010.
Published: February 2011.