Fast Incremental Planarity Testing

The incremental planarity testing problem is to perform the following operations on a biconnected planar graph G of at most n vertices: test if an edge can be added between two vertices while preserving planarity; add edges and vertices that preserve planarity. Let m be the total number of operations. We present fast data structures for this problem that can be used in conjunction with the previous algorithm of Di Battista and Tamassia to achieve an O((m; n)) worst-case amortized time per test operation. If the graph is bicon-nected, a sequence of n additions can be performed in total time O(mm(m;n)) worst-case plus O(n) expected time. Our tree data structure is exible and can answer in O(1) time queries about parents, roots, and nearest common ancestors while performing tree modiications such as inserting nodes, cutting edges, and merging or splitting nodes. If the graph is not biconnected then insertions of edges and vertices require O(log n) amortized expected time per operation. The study of graph planarity has a long and rich history. The oo-line planarity testing problem, in which one must test if a given graph is planar, has been studied since the nineteen-sixties. If can be solved in O(n) sequential time 3, 12] and O(log n) time on a CRCW PRAM 17]. Recently Di Battista and Tamassia introduced and studied the incremental planarity testing problem 1, 2]: beginning with a biconnected graph G process on-line an intermingled sequence of the following query and update operations. 1. Test(u; v): Determine if an edge can be inserted between vertices u and v of while still preserving the planarity of G. 2. Insert Vertex(v; e): Split edge e into two edges e 1 , e 2 by inserting a new vertex v. Return e 1 and e 2. 3. Insert Edge(u; v): Insert a new edge e between u and v. Return e. The insertion must preserve planarity. The above repertoire of operations is suucient to construct any biconnected pla-nar graph. A related problem is static on-line testing problem, in which one must preprocess a xed planar graph G so as to answer on-line a sequence of Test(u; v) operations. Di Battista and Tamassia gave an algorithm that performs Test(u; v) in