Fast incremental planarity testing

Abstract

The incremental planarity testin 9 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(oα(m, n)) worst-case amortized time per test operation. If the graph is biconnected, a sequence of n additions can be performed in total time O(mα(m, n)) worst-case plus O(n) expected time. Our tree data structure is flexible and can answer in 0(1) time queries about parents, roots, and nearest common ancestors while performing tree modifications 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.