Planar Separators and the Euclidean Norm

In this paper we show that every 2-connected embedded planar graph with faces of sizes d! . . . . . df has a simple cycle separator of size 1.58~,/dl 2 + . . . + d } and we give an almost linear time algorithm for finding these separators, O(m~(n. n)). We show that the new upper bound expressed as a function of IG{ = ~/d~ +..+ d} is no larger, up to a constant factor than previous bounds that where expressed in terms of ~ S . v where d is the maximum face size and / v is the number of vertices and is much smaller for many graphs. The algorithms developed are simpler than earlier algorithms in that they work directly with the planar graph and its dual. They need not construct or work with the face-incidence graph as in [Mi186, GM87, GM]. 1 I n t r o d u c t i o n Planar graphs have played an important role in both sequential as well as parallel algorithm design. They arise in may areas of computation including: numerical analysis, animation, and VLSI. One of the important properties possessed by planar graphs, but not true for general graphs, is that they have small separators. Historically, a separator, in a graph G = (V, E), is a subset C C V such that (1) the remaining vertices can be partitioned into two sets: A and B, (2) IAI. IB{ ~ 2/3{VI, and (3) there are no edges between vertices in A and vertices in B. Lipton and Tarjan were the first to show that paper graphs have O(,fv) separators, [LT79]. Improvements in the constant have been made by Djidjev and the first author, [Dji81, Gaz86]. There are two important additional properties we require of separators. First, we shall assign a weighting function # to the vertices, edges, and faces of the embedded planar graph and require that the separator will "separate" the weighted graph. Second, we shall require that the separator be a cycle or collection of cycles. Intuitively, this means that we separate the planar graph "drawn" on the plane by cutting along the edges of the graph. Since all the separators we construct in this paper are in fact simple cycles, we will restrict our attention to the simple cycle case. These two restrictions have been addressed by the second author in [Mi186]. We will assume that the reader has some knowledge of this paper. The following definition is from [Mi186]. ~This work was supported in part by grant number N00014-88-K-0623 iT his work was supported in part by National Science Foundation grant DCR-8713489.