Undirected Connectivity of Sparse Yao Graphs

TL;DR

This paper investigates the connectivity of Yao subgraphs derived from disk graphs in the plane, establishing bounds on the radius needed for connectivity for different numbers of sectors, relevant for directional antenna networks.

Contribution

It provides new bounds on the minimum radius for connectivity of Yao subgraphs with small sector counts, advancing understanding of their properties in geometric network models.

Findings

Y_4[G^d] is connected if and only if d >= sqrt(2).

Y_3[G^d] can be disconnected for d <= 1.056 and always connected for d >= 2/sqrt(3).

Y_2[G^d] can be disconnected for any d >= 1.

Abstract

Given a finite set S of points in the plane and a real value d > 0, the d-radius disk graph G^d contains all edges connecting pairs of points in S that are within distance d of each other. For a given graph G with vertex set S, the Yao subgraph Y_k[G] with integer parameter k > 0 contains, for each point p in S, a shortest edge pq from G (if any) in each of the k sectors defined by k equally-spaced rays with origin p. Motivated by communication issues in mobile networks with directional antennas, we study the connectivity properties of Y_k[G^d], for small values of k and d. In particular, we derive lower and upper bounds on the minimum radius d that renders Y_k[G^d] connected, relative to the unit radius assumed to render G^d connected. We show that d=sqrt(2) is necessary and sufficient for the connectivity of Y_4[G^d]. We also show that, for d <= ~1.056, the graph Y_3[G^d] can be disconnected, but for d >= 2/sqrt(3), Y_3[G^d] is always connected. Finally, we show that Y_2[G^d] can be disconnected, for any d >= 1.