Spanning Properties of Theta-Theta-6

TL;DR

This paper investigates the spanning properties of the Theta-Theta-6 graph, showing it is a spanner for convex point sets but not for non-convex sets, highlighting its limitations and strengths.

Contribution

It proves that Theta-Theta-6 is a spanner for convex point sets and demonstrates its unbounded spanning ratio for non-convex sets.

Findings

Theta-Theta-6 is a spanner for convex point sets.

The spanning ratio of Theta-Theta-6 is unbounded for non-convex sets.

Contrast with Yao-Yao graph YY_6, which is not a spanner in this context.

Abstract

We show that, unlike the Yao-Yao graph $YY_6$, the Theta-Theta graph $\Theta\Theta_6$ defined by six cones is a spanner for sets of points in convex position. We also show that, for sets of points in non-convex position, the spanning ratio of $\Theta\Theta_6$ is unbounded.