# Complexity of Planar Embeddability of Trees inside Simple Polygons

## TL;DR

This paper proves that deciding if a tree can be embedded inside a simple polygon without crossings is NP-complete, highlighting the computational difficulty of geometric graph embedding problems.

## Contribution

It establishes the NP-completeness of planar straight-line tree embedding within simple polygons, resolving an open problem in geometric graph theory.

## Key findings

- Embedding trees inside simple polygons is NP-complete.
- Straight-line constrained point-set embedding of trees is NP-complete.
- Addresses an open problem in geometric graph embedding complexity.

## Abstract

Geometric embedding of graphs in a point set in the plane is a well known problem. In this paper, the complexity of a variant of this problem, where the point set is bounded by a simple polygon, is considered. Given a point set in the plane bounded by a simple polygon and a free tree, we show that deciding whether there is a planar straight-line embedding of the tree on the point set inside the simple polygon is NP-complete. This implies that the straight-line constrained point-set embedding of trees is also NP-complete, which was posed as an open problem in [8].