# Pseudo-Triangle Visibility Graph: Characterization and Reconstruction

## TL;DR

This paper introduces a polynomial characterization and a linear time reconstruction algorithm for pseudo-triangle visibility graphs, expanding understanding of polygon visibility relations beyond special cases.

## Contribution

It provides the first polynomial characterization and a linear time reconstruction algorithm for pseudo-triangle visibility graphs, generalizing previous special case solutions.

## Key findings

- A set of necessary and sufficient properties for pseudo-triangle visibility graphs.
- A polynomial-time algorithm to verify if a graph is a pseudo-triangle visibility graph.
- A linear-time algorithm for reconstructing a pseudo-triangle from its visibility graph.

## Abstract

The visibility graph of a simple polygon represents visibility relations between its vertices. Knowing the correct order of the vertices around the boundary of a polygon and its visibility graph, it is an open problem to locate the vertices in a plane in such a way that it will be consistent with this visibility graph. This problem has been solved for special cases when we know that the target polygon is a {\it tower} or a {\it spiral}. Knowing that a given visibility graph belongs to a simple polygon with at most three concave chains on its boundary, a {\it pseuodo-triangle}, we propose a linear time algorithm for reconstructing one of its corresponding polygons. Moreover, we introduce a set of necessary and sufficient properties for characterizing visibility graphs of pseudo-triangles and propose polynomial algorithms for checking these properties.