On Triangulating Planar Graphs under the Four-Connectivity Constraint

Abstract

Triangulation of planar graphs under constraints is a fundamental problem in the representation of objects. Related keywords are graph augmentation from the field of graph algorithms and mesh generation from the field of computational geometry. We consider the triangulation problem for planar graphs under the constraint to satisfy 4-connectivity. A 4-connected planar graph has no separating triangles, i.e., cycles of length 3 which are not a face.We show that triangulating embedded planar graphs without introducing new separating triangles can be solved in linear time and space. If the initial graph had no separating triangle, the resulting triangulation is 4-connected. If the planar graph is not embedded, then deciding whether there exists an embedding with at most k separating triangles is NP-complete. For biconnected graphs a linear-time approximation which produces an embedding with at most twice the optimal number is presented. With this algorithm we can check in linear time whether a biconnected planar graph can be made 4-connected while maintaining planarity. Several related remarks and results are included. Introduction. The problem of augmenting a graph to reach a certain connectivity requirement by adding edges has important applications in network reliability [9], [25]and fault-tolerant computing. The general version of the augmentation problem is to augment the input graph to reach a given connectivity requirement by adding a smallest set of edges. Recent papers present linear-time augmentation algorithms to admit the 2-connectivity constraint [8], [24], [15], and the 3-connectivity constraint [14]. With respect to 4-connectivity Kanevsky et al. [16] presented an O(n · α(m, n) + m)-time algorithm for testing 4-connectivity, and Hsu presented an O(n · α(m, n) + m)-time algorithm to compute the minimal set of edges to augment a 3-connected graph to a 4-connected graph [13] (here α(m, n) is the functional inverse of Ackermann's function). Kant described several algorithms for the augmentation problem with the additional constraint of planarity [18]. This problem has important applications in planar network design and graph-drawing algorithms.

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