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 .
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.
@cite_4 @cite_3 @cite_11
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We consider graphs that admit polyline drawings where all crossings occur at the same angle @math . We prove that every graph on @math vertices that admits such a polyline drawing with at most two bends per edge has @math edges. This result remains true when each crossing occurs at an angle from a small set of angles. We also provide several extensions that might be of independent interest. In this paper, we study the problem of drawing a given planar graph such that vertices are at pre-specified points and the entire drawing is inside a given polygon. We give a method that shows that for an n-vertex graph and a k-sided polygon, Θ(kn2) bends are always sufficient. We also give an example of a graph where Θ(kn2) bends are necessary for such a drawing. Abstract We study planar orthogonal drawings of graphs and provide lower bounds on the number of bends along the edges. We exhibit graphs on n vertices that require Ω( n ) bends in any layout, and show that there exist optimal drawings that require Ω( n ) bends and have all of them on a single edge of length Ω( n 2 ). This work finds applications in VLSI layout, aesthetic graph drawing, and communication by light or microwave. Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A geometric point-set embedding is a point-set embedding with no edge bends. This paper studies the following problem: The input is a set S of n points, a planar graph G with n vertices, and a geometric point-set embedding of a subgraph G^'@?G on a subset of S. The desired output is a point-set embedding of G on S that includes the given partial drawing of G^'. We concentrate on trees and show how to compute the output in O(n^2logn) time in a real-RAM model and with at most n-k edges with at most [email protected]?k [email protected]? bends, where k is the number of vertices of the given subdrawing. We also prove that there are instances of the problem which require at least k-3 bends on n-k edges.