Undirected connectivity of sparse Yao graphs

  • Authors:
  • Mirela Damian;Abhaykumar Kumbhar

  • Affiliations:
  • Villanova University, Villanova, PA;Villanova University, Villanova, PA

  • Venue:
  • FOMC '11 Proceedings of the 7th ACM ACM SIGACT/SIGMOBILE International Workshop on Foundations of Mobile Computing
  • Year:
  • 2011

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Abstract

Given a finite set S of points in the plane and a real value d 0, the d--radius disk graph Gd 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 Yk[G] with integer parameter k 0 contains, for each point p ∈ S, a shortest edge pq ∈ 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 Yk[Gd], for small values of k and d. In particular, we derive lower and upper bounds on the minimum radius d that renders Yk[Gd] connected, relative to the unit radius assumed to render Gd connected. We show that d = [EQUATION] is necessary and sufficient for the connectivity of Y4[Gd]. We also show that, for d ≤ 5 − 2/3 [EQUATION], the graph Y3[Gd] can be disconnected, but Y3[G2/[EQUATION]] is always connected. Finally, we show that Y2[Gd] can be disconnected, for any d ≥ 1.