text: The thickness of a graph @math with @math vertices is the minimum number of planar subgraphs of @math whose union is @math . A polyline drawing of @math in @math is a drawing @math of @math , where each vertex is mapped to a point and each edge is mapped to a polygonal chain. Bend and layer complexities are two important aesthetics of such a drawing. The bend complexity of @math is the maximum number of bends per edge in @math , and the layer complexity of @math is the minimum integer @math such that the set of polygonal chains in @math can be partitioned into @math disjoint sets, where each set corresponds to a planar polyline drawing. Let @math be a graph of thickness @math . By F ' a ry's theorem, if @math , then @math can be drawn on a single layer with bend complexity @math . A few extensions to higher thickness are known, e.g., if @math (resp., @math ), then @math can be drawn on @math layers with bend complexity 2 (resp., @math ). However, allowing a higher number of layers may reduce the bend complexity, e.g., complete graphs require @math layers to be drawn using 0 bends per edge. In this paper we present an elegant extension of F ' a ry's theorem to draw graphs of thickness @math . We first prove that thickness- @math graphs can be drawn on @math layers with @math bends per edge. We then develop another technique to draw thickness- @math graphs on @math layers with bend complexity, i.e., @math , where @math . Previously, the bend complexity was not known to be sublinear for @math . Finally, we show that graphs with linear arboricity @math can be drawn on @math layers with bend complexity @math .
  @cite_4: 
  @cite_3: Let G be a planar graph with n vertices and with a partition of the vertex set into subsets V"0,...,V"k"-"1 for some positive integer [email protected][email protected]?n. Let S be a set of n distinct points in the plane with a partition into subsets S"0,...,S"k"-"1 with |V"i|=|S"i| ([email protected][email protected]?k-1). This paper studies the problem of computing a planar polyline drawing of G, such that each vertex of V"i is mapped to a distinct point of S"i. Lower and upper bounds on the number of bends per edge are proved for any [email protected][email protected]?n. In the special case k=n, we improve the upper and lower bounds presented in a paper by Pach and Wenger [J. Pach, R. Wenger, Embedding planar graphs at fixed vertex locations, Graphs and Combinatorics 17 (2001) 717-728]. The upper bound is based on an algorithm for computing a topological book embedding of a planar graph, such that the vertices follow a given left-to-right order and the number of crossings between every edge and the spine is asymptotically optimal, which can be regarded as a result of independent interest.
  @cite_11: Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices. This forms the basis for the visualization of dynamic graphs and thus is an important field of research. Recently there has been a great deal of work investigating simultaneous embedding problems both from a theoretical and a practical point of view. We survey recent work on this topic.
summary: Given an @math -vertex planar graph @math and a point location for each vertex in @math , Pach and Wenger @cite_11 showed that @math admits a planar polyline drawing with the given vertex locations, where each edge has at most @math bends. They also showed that @math bends are sometimes necessary. @cite_3 and Gordon @cite_4 independently improved the bend complexity to @math . Consequently, for @math , these constructions can be used to draw @math on @math layers with at most @math bends per edge.