Spanning Properties of Theta-Theta Graphs

TL;DR

This paper investigates the spanning properties of Theta-Theta graphs, establishing new bounds on their spanning ratios and demonstrating their effectiveness as geometric spanners.

Contribution

It provides the first known spanning ratio bounds for Theta-Theta graphs, improving previous analyses and extending the class of such graphs with new ratio guarantees.

Findings

Spanning ratio of 7.82 for Theta-Theta graphs with certain parameters.

Spanning ratio of 16.76 for extended Theta-Theta graphs.

Spanning ratio decreases to 4.64 as parameter k' increases to 8.

Abstract

We study the spanning properties of Theta-Theta graphs. Similar in spirit with the Yao-Yao graphs, Theta-Theta graphs partition the space around each vertex into a set of k cones, for some fixed integer k > 1, and select at most one edge per cone. The difference is in the way edges are selected. Yao-Yao graphs select an edge of minimum length, whereas Theta-Theta graphs select an edge of minimum orthogonal projection onto the cone bisector. It has been established that the Yao-Yao graphs with parameter k = 6k' have spanning ratio 11.67, for k' >= 6. In this paper we establish a first spanning ratio of $7.82$ for Theta-Theta graphs, for the same values of $k$. We also extend the class of Theta-Theta spanners with parameter 6k', and establish a spanning ratio of $16.76$ for k' >= 5. We surmise that these stronger results are mainly due to a tighter analysis in this paper, rather than Theta-Theta being superior to Yao-Yao as a spanner. We also show that the spanning ratio of Theta-Theta graphs decreases to 4.64 as k' increases to 8. These are the first results on the spanning properties of Theta-Theta graphs.