Collective Tree Spanners for Unit Disk Graphs with Applications Feodor F. Dragan, Yang Xiang and Chenyu Yan Algorithmic Research Laboratory, Department of Computer Science Kent State University, Kent, Ohio, U.S.A. {cyan,yxiang,dragan}@cs.kent.edu Abstract In this paper, we establish a novel balanced separator theorem for Unit Disk Graphs (UDGs), which mimics the well-known Lipton and Tarjan’s planar balanced shortest paths separator theorem. We prove that, in any n-vertex UDG G, one can find two hop-shortest paths P (s, x) and P (s, y) such that the removal of the 3-hop- neighborhood of these paths (i.e., N 3 G [P (s, x) ∪ P (s, y)]) from G leaves no connected component with more than 2/3n vertices. This new balanced shortest-paths—3- hop-neighborhood separator theorem allows us to build, for any n-vertex UDG G, a system T (G) of at most 2 log 3 2 n + 2 spanning trees of G such that, for any two vertices x and y of G, there exists a tree T in T (G) with d T (x, y) ≤ 3 · d G (x, y)+ 12. That is, the distances in any UDG can be approximately represented by the distances in at most 2 log 3 2 n + 2 of its spanning trees. Using these results, we propose a new compact and low delay routing labeling scheme for UDGs. Keywords: unit disk graphs, collective tree spanners, routing and distance labeling schemes, balanced separators, efficient graph algorithms. Electronic Notes in Discrete Mathematics 32 (2009) 117–124 1571-0653/$ – see front matter, Published by Elsevier B.V. www.elsevier.com/locate/endm doi:10.1016/j.endm.2009.02.016