# On Plane Constrained Bounded-Degree Spanners

## TL;DR

This paper demonstrates that the constrained half-θ6-graph is a plane 2-spanner of the visibility graph and introduces a method to construct a plane 6-spanner with bounded degree, improving geometric network efficiency.

## Contribution

It proves the constrained half-θ6-graph is a plane 2-spanner of the visibility graph and presents a construction for a plane 6-spanner with bounded maximum degree.

## Key findings

- Constrained half-θ6-graph is a plane 2-spanner of the visibility graph.
- A method to construct a plane 6-spanner with maximum degree 6+c.
- The spanner maintains planarity and bounded degree properties.

## Abstract

Let $P$ be a finite set of points in the plane and $S$ a set of non-crossing line segments with endpoints in $P$. The visibility graph of $P$ with respect to $S$, denoted $Vis(P,S)$, has vertex set $P$ and an edge for each pair of vertices $u,v$ in $P$ for which no line segment of $S$ properly intersects $uv$. We show that the constrained half-$\theta_6$-graph (which is identical to the constrained Delaunay graph whose empty visible region is an equilateral triangle) is a plane 2-spanner of $Vis(P,S)$. We then show how to construct a plane 6-spanner of $Vis(P,S)$ with maximum degree $6+c$, where $c$ is the maximum number of segments of $S$ incident to a vertex.