Geometric Simultaneous Embeddings of a Graph and a Matching

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 Article type: Regular paper Communicated by: D. Eppstein and E. R. Gansner E-mail addresses: [email protected] (Sergio Cabello) [email protected] (Marc van Kreveld) [email protected] (Giuseppe Liotta) [email protected] (Henk Meijer) [email protected] (Bettina Speckmann) [email protected] (Kevin Verbeek) 80 Cabello et al. Geom. Simult. Embeddings of a Graph and a Matching