The geometric simultaneous embedding problem asks whether two pla- nar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous em- bedding 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 strength- ens an analogous negative result for a planar graph and a path. On the positive side, we describe algorithms that compute a geometric simulta- neous 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.

### Geometric Simultaneous Embeddings of a Graph and a Matching

#### Abstract

The geometric simultaneous embedding problem asks whether two pla- nar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous em- bedding 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 strength- ens an analogous negative result for a planar graph and a path. On the positive side, we describe algorithms that compute a geometric simulta- neous 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.
##### Scheda breve Scheda completa Scheda completa (DC)
2011
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11391/282894`
• ND
• ND
• ND