Recognition of Triangulation Duals of Simple Polygons With and Without Holes

TL;DR

This paper studies the problem of recognizing whether a given graph is the dual of a triangulation of a simple polygon, analyzing how the problem's complexity varies with different information levels and identifying tractable and intractable cases.

Contribution

It provides a detailed complexity analysis of the graph recognition problem for polygon triangulations, establishing clear boundaries between solvable and hard instances.

Findings

Identifies conditions under which the recognition problem is tractable.

Establishes intractability results for certain versions of the problem.

Provides a complexity boundary based on available information.

Abstract

We investigate the problem of determining if a given graph corresponds to the dual of a triangulation of a simple polygon. This is a graph recognition problem, where in our particular case we wish to recognize a graph which corresponds to the dual of a triangulation of a simple polygon with or without holes and interior points. We show that the difficulty of this problem depends critically on the amount of information given and we give a sharp boundary between the various tractable and intractable versions of the problem.