Incremental Convex Planarity Testing

• Published in: Information and computation Vol. 169; no. 1; pp. 94 - 126
• Main Authors: Di Battista, Giuseppe, Tamassia, Roberto, Vismara, Luca
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 25.08.2001; Elsevier
• Subjects: Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Graph theory; Mathematics; Sciences and techniques of general use; Graph theory; VLSI; Test; Planarity testing; Planar graph; Complexity
• ISSN: 0890-5401; 1090-2651
• DOI: 10.1006/inco.2001.3031

An 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).