Cognitive measurements of graph aesthetics
Information Visualization
On simultaneous planar graph embeddings
Computational Geometry: Theory and Applications
Matched drawings of planar graphs
GD'07 Proceedings of the 15th international conference on Graph drawing
Constrained simultaneous and near-simultaneous embeddings
GD'07 Proceedings of the 15th international conference on Graph drawing
Simultaneous geometric graph embeddings
GD'07 Proceedings of the 15th international conference on Graph drawing
Matched drawability of graph pairs and of graph triples
Computational Geometry: Theory and Applications
On a tree and a path with no geometric simultaneous embedding
GD'10 Proceedings of the 18th international conference on Graph drawing
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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 exists a planar graph and a matching which do not admit a geometric simultaneous embedding. This generalizes the same 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 proof for a matching and a tree sheds new light on a major open question: do a tree and a path always admit a geometric simultaneous embedding? 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.