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C. et fApidMteaisadSaitc,SoyBokUn Brook Stony Statistics, and Mathematics Applied of Dept. et fCmue cec,Ui.o Miami, of Univ. Science, Computer of Dept. nSmlaeu rp Embedding Graph Simultaneous On 2 ecnie h rbe fsmlaeu medn fpla- of embedding simultaneous of problem the consider We et fCmue cec,Ui.o Arizona, of Univ. Science, Computer of Dept. O { alon,erten,kobourov ( 1 n .Efrat A. , ) × O ( [email protected] n .S .Mitchell B. S. J. rd n notrlnradgnrlplanar general and outerplanar an and grid, ) ∗ O 2 .Erten C. , ( n 2 ) 1 × O } @cs.arizona.edu ( n ∗ 3 2 grid. ) ⋆⋆ .G Kobourov G. S. , 3 O ( [email protected] n 2 ) × O ( Foundation. n n 3 ac (NAG2-1325), earch rd fthe If grid. ) ps nti area, this In aphs. hs ihol a only with those × ⋆ n 2 iversity, rd or grid, ftm and time of e and , sadedges, and es multa- hthe ch culdraw- actual a s slight is, hat rab itself by eria thout other atclap- ractical mentations pe and epted e hand, her n infor- ing ignificant locations great a is From a theoretical point-of-view, one recent trend in research has been to address the topic of thickness of graphs [10]. The thickness of a graph is the minimum number of planar subgraphs into which the graph can be partitioned. Similar to a common technique in VLSI design, the graph is embedded in layers. Any two edges drawn in the same layer can only intersect at a common vertex, and vertices are placed are placed in the same location across all layers. Thus, the property of graph thickness formalizes the notion of layered drawing of graphs. The thickness of a graph is defined as the minimum number of layers for which a drawing of G exists, in which edges can be drawn as Jordan curves [1]. A related graph property is geometric thickness and defined as the minimum number of layers for which a drawing of G exists, in which all edges are drawn as straight-line segments [9]. Finally, book thickness of a graph is the minimum number of layers for which a drawing of G exists, in which edges are drawn as straight-line segments and vertices are in convex position [3]. We look at this problem almost in reverse. Assume we are given the layered subgraphs and now wish to embed the various layers so that the vertices coin- cide and so that no edges cross. We wish to simultaneously draw the layered subgraphs, that is, we wish to display several graphs using the same set of ver- tices but different sets of edges. Take, for example, two graphs from the 1998 Worldcup; see Fig. 1. One of the graphs is a tree illustrating the games played. The other is a graph showing the major exporters and importers of players on the club level. In displaying the information, one could certainly look at the two graphs separately, but then there would be little correspondence between the two layouts if they were created independently, since the viewer has no “mental map” between the two graphs. Using a simultaneous embedding, the vertices could be placed in the exact same locations for both graphs and the relation- ships become more clear. This is different than simply merging the two graphs together and displaying the information as one large graph. In simultaneous embeddings, we are concerned with crossings but not be- tween edges belonging to different layers (and thus different graphs). In the merged version, the graph drawing algorithm used loses all information about the separation of the two graphs and so must also avoid such non-essential cross- ings. The techniques for displaying simultaneous embeddings are also quite var- ied. One may choose to draw all graphs simultaneously, employing different edge styles, colors, and thickness for each edge set. One may choose a more three- dimensional approach to giving a visual difference between layers. One may also choose to show only one graph at a time and allow the users to choose which graph they wish to see; however, the vertices would not move, only the edge sets would change. Finally, one may highlight one set of edges over another giving the effect of bolding certain graphs. The subject of simultaneous embeddings has many different variants, several of which we address here. The two main classifications we consider are embed- dings with (and without) predefined vertex mappings.
– Simultaneous (Geometric) Embeddings With Mapping: Given k pla- nar graphs Gi(V, Ei) for 1 i k, find plane straight-line drawings Di of ≤ ≤
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