We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem---to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph---and the simultaneous planarity problem---to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required planar drawings exist, then there are planar drawings with a linear number of bends per edge and, in the case of simultaneous planarity, a constant number of crossings between every pair of edges. Our proofs provide efficient algorithms if the combinatorial embedding of the drawing is given. Our result on partially embedded graph drawing generalizes a classic result by Pach and Wenger which shows that any planar graph can be drawn with a linear number of bends per edge if the location of each vertex is fixed.
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. 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. 