On-Line Convex Plabarity Testing

An important class of planar straight-line drawings of graphs are the convex drawings, in which all faces are drawn as convex polygons. A graph is said to be convex planar if it admits a convex drawing. We consider the problem of testing convex planarity in a semidynamic environment, where a graph is subject to on-line insertions of vertices and edges. We present on-line algorithms for convex planarity testing with the following performance, where n denotes the number of vertices of the graph: convex planarity testing and insertion of vertices take O(1) worst-case time, insertion of edges takes O(log n) amortized time, and the space requirement of the data structure is O(n). Furthermore, we give a new combinatorial characterization of convex planar graphs.