Undirected connectivity of sparse Yao graphs

Given a finite set <i>S</i> of points in the plane and a real value <i>d</i> > 0, the <i>d</i>--radius disk graph <i>G</i>^<i>d</i> contains all edges connecting pairs of points in <i>S</i> that are within distance <i>d</i> of each other. For a given graph <i>G</i> with vertex set <i>S</i>, the Yao subgraph <i>Y</i>_<i>k</i>[<i>G</i>] with integer parameter <i>k</i> > 0 contains, for each point <i>p</i> ∈ <i>S</i>, a shortest edge <i>pq</i> ∈ <i>G</i> (if any) in each of the <i>k</i> sectors defined by <i>k</i> equally-spaced rays with origin <i>p.</i> Motivated by communication issues in mobile networks with directional antennas, we study the connectivity properties of <i>Y</i>_<i>k</i>[<i>G</i>^<i>d</i>], for small values of <i>k</i> and <i>d.</i> In particular, we derive lower and upper bounds on the minimum radius <i>d</i> that renders <i>Y</i>_<i>k</i>[<i>G</i>^<i>d</i>] connected, relative to the unit radius assumed to render <i>G</i>^<i>d</i> connected. We show that <i>d</i> = [EQUATION] is necessary and sufficient for the connectivity of <i>Y</i>₄[<i>G</i>^<i>d</i>]. We also show that, for <i>d</i> ≤ 5 − 2/3 [EQUATION], the graph <i>Y</i>₃[<i>G</i>^<i>d</i>] can be disconnected, but <i>Y</i>₃[<i>G</i>^2/[EQUATION]] is always connected. Finally, we show that <i>Y</i>₂[<i>G</i>^<i>d</i>] can be disconnected, for any <i>d</i> ≥ 1.