Balanced line separators of unit disk graphs

Practical Compact Routing on Sparse Graphs

“…The celebrated theorem of Lipton and Tarjan [2] guarantees that every n-vertex planar graph contains a separator of size O( √ n), such that the size of each separated component is between n/3 and 2n/3. Similar theorems exist also for UDG's [1]. A simple and common way that we use to generate good separators, which usually give components with this property, is the spectral method [21], where the socalled Fiedler eigenvector of the graph Laplacian matrix is used.…”

“…This enables an O(log n) spanning tree cover, with stretch 3. Improvement on separator theorems for UDG's have been obtained over the years, culminating in the results of Carmi et al [1], who also prove a bound on the size of the separator. Kaplan et al [12] describe a routing method for UDG's based on the existence of a well-separated pair decomposition of the graph and modifiable headers with a polylogarithmic number of bits.…”

“…Sparse graphs such as UDG's and plane graphs have an important property that we will take advantage of: the existence of compact balanced separators [1], [2]. This means that given such a graph G with vertex set V of size n, there always exists a subset of the vertices S of size O( √ n), such that removal of these vertices and the edges incident on them separates the graph into two disjoint components with vertex sets U 0 and U 1 , both of size at least n/3.…”

Space-Efficient Algorithms for Reachability in Geometric Graphs

“…The vertices of a penny graph are unit circles in the plane, such that no two of those circles cross each other, and there is an edge between two vertices if and only if the corresponding circles touches each other. Due to Carmi et al [7], we know that given a set of unit disks, there exists an axis-parallel line intersecting O( √ m + n) disks such that each half-plane separated by that line contains at most 4n/5 disks. They call such a separator a balanced separator.…”

“…|C|, and no element of C 1 has a point in common with any element of C 2 . Recently, Carmi et al [7] improved the bound for unit disk graphs. They showed for any set of n unit disks in the plane there exists a line such that intersects at most O( (m + n) log n) disks and each of the halfplanes determined by contains at most 2n/3 unit disks from the set, where m is the number of intersecting pairs of disks.…”