# Folding a triangulated simple polygon: Structural and algorithmic results

## Abstract

We describe a linear-time algorithm that folds a triangulated simple polygon into a single triangle. Using this technique, we derive a particularly simple proof of Chvàtal's art gallery theorem and improve or simplify a number of algorithms that deal with triangulated simple polygons. We describe two improved algorithms, both based on the degree sequence of the boundary vertices of the given triangulated simple polygon. The first is a linear-time algorithm that, using only the parity of the degree of each vertex, colors the vertices of a triangulated simple polygon using three colors. The second algorithm reconstructs the polygon and its triangulation from the degree sequence. We then show that our results extend to *k*-trees.

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