Parallel Transitive Closure and Point Location in Planar Structures

AMS(MOS) subject classifications. 68E05, 68C05, 68C25Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar st-graphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of these algorithms achieve optimal O(logn) running time using n/logn processors in the EREW PRAM model, n being the number of vertices

A Dynamic Data Structure for Planar Graph Embedding

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryNational Science Foundation / ECS-84-1090

Incremental Convex Planarity Testing

AbstractAn important class of planar straight-line drawings of graphs are convex drawings, in which all the faces are drawn as convex polygons. A planar graph is said to be convex planar if it admits a convex drawing. We give a new combinatorial characterization of convex planar graphs based on the decomposition of a biconnected graph into its triconnected components. We then consider the problem of testing convex planarity in an incremental environment, where a biconnected planar graph is subject to on-line insertions of vertices and edges. We present a data structure for the on-line incremental convex planarity testing problem with the following performance, where n denotes the current number of vertices of the graph: (strictly) convex planarity testing takes O(1) worst-case time, insertion of vertices takes O(log n) worst-case time, insertion of edges takes O(log n) amortized time, and the space requirement of the data structure is O(n)