# On triangulating planar graphs under the four-connectivity constraint

## Abstract

Triangulation under constraints is a fundamental problem in the representation of objects. Graph augmentation and mesh generation are related keywords from the areas of graph algorithms and computational geometry. In this paper we consider the triangulation problem for planar graphs under the constraint that four-connectivity has to be satisfied.

Our first result states that triangulating embedded planar graphs without introducing new separating triangles can be solved in linear time and space. If the planar graph is not embedded then deciding whether there exists an embedding with at most *K* separating triangles is NP-complete. A linear time algorithm for this problem is presented, yielding a solution with at most twice the optimal number. Several related remarks and results are included as well.

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