Additional PC-Tree Planarity Conditions

Abstract

Abstract. Recent research efforts have produced new algorithms for solving planarity-related problems. One such method performs vertex addition using the PC-tree data structure, which is similar to but simpler than the well-known PQ-tree. For each vertex, the PC-tree is first checked to see if the new vertex can be added without violating certain planarity conditions; if the conditions hold, the PC-tree is adjusted to add the new vertex and processing continues. The full set of planarity conditions are required for a PC-tree planarity tester to report only planar graphs as planar. This paper provides further analyses and new planarity conditions needed to produce a correct planarity algorithm with a PC-tree.

On the Cutting Edge: Simplified O(n) Planarity by Edge Addition

“…Our edge addition method is presented using only graph constructs, but our graph theoretic analysis of correctness is applicable to the underlying graph represented by a PC-tree. Thus, our proof of correctness justifies the application of our definitions, path traversal process and edge embedding technique to substantially redesign the PC-tree processing, eliminating the numerous planarity conditions identified in [3,23].…”

“…The reduction has the effect of embedding all edges from v to its DFS descendants while maintaining planarity. To be sure that the reduction maintains planarity, a correct PC-tree algorithm must test the planarity conditions in [23] as well as the additional conditions identified in [3].…”

“…The expressed purpose of both PC-tree planarity and our own vertex addition method work was to present a simplified linear time planarity algorithm relative to the early achievements of Hopcroft and Tarjan [12] and Booth and Lueker [1]. While it is often accepted that the newer algorithms in [2] and [23] are simpler, the corresponding expositions in [5] and [3] clearly show that there is still complexity in the details. This additional complexity results from a 'batch' approach of vertex addition, in which back edges from a vertex to its descendants are embedded only after a set of planarity condition tests have been performed.…”

Errors in graph embedding algorithms

“…3. The edge [1,6] is a repeated path on the facial boundary. The endpoints of this path are connected by paths a = [1, 4, 3, 2, 1] and b = [6,3,4,5,6] which are part of the facial boundary, and which are cycles in the graph.…”

“…The planarity testing algorithm of Shih and Hsu [25] requires clarification in order to provide an implementable version. Boyer [1] presented the additional planarity conditions required to program it using the PC-tree data structure described in [25]. A projective plane embedding algorithm derived by Perunicic and Duric [24] is incorrect in that it sometimes fails to find a projective plane embedding of a graph when one exists as noted by Mohar [18, p. 483] and independently observed by Williamson (private communication to Mohar [18,p.…”