Abstract. A collection of n balls in d dimensions forms a k-ply system if no point in the space is covered by more than k balls. We show that for every k-ply system �, there is a sphere S that intersects at most O(k 1/d n 1�1/d) balls of � and divides the remainder of � into two parts: those in the interior and those in the exterior of the sphere S, respectively, so that the larger part contains at most (1 � 1/(d � 2))n balls. This bound of O(k 1/d n 1�1/d) is the best possible in both n and k. We also present a simple randomized algorithm to find such a sphere in O(n) time. Our result implies that every k-nearest neighbor graphs of n points in d dimensions has a separator of size O(k 1/d n 1�1/d). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk-packing, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.

Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph together with its ends. Our approach

The adaption of combinatorial duality to infinite graphs has been hampered by the fact that while cuts (or cocycles) can be infinite, cycles are finite. We show that these obstructions fall away when duality is reinterpreted on the basis of a ‘singular’ approach to graph homology, whose cycles are defined topologically in a space formed by the graph together with its ends and can be infinite. Our approach enables us to complete Thomassen’s results about ‘finitary ’ duality for infinite graphs to full duality, including his extensions of Whitney’s theorem.

We address in this paper the problem of constructing embeddings of planar graphs satisfying declarative, user-defined topological constraints. The constraints consist each of a cycle...

This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. This approach has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. The shift of paradigm it proposes is thus as much an answer to old questions as a source of new ones; many concrete problems of both types are suggested in the paper. This paper attempts to provide an entry point to this field for readers that have not followed the literature that has emerged in the last 10 years or so. It takes them on a quick route through what appear to be the most important lasting results, introduces them to key proof techniques, identifies the most promising open

. A plane integral drawing of a planar graph G is a realization of G in the plane such that the vertices of G are mapped into distinct points and the edges of G are mapped into straight line segments of integer length which connect the corresponding vertices such that two edges have no inner point in common. We conjecture that plane integral drawings exist for all planar graphs, and we give parts of a proof of this conjecture. 1. Introduction A plane integral drawing of a planar graph G is a realization of G in the plane such that the vertices of G are mapped into distinct points, also called vertices, and the edges of G are mapped into straight line segments of integer length, also called edges, which connect the corresponding vertices such that two edges have no inner point in common. The existence of a plane drawing with straight line segments for all edges is known for every planar graph (Wagner, F' ary [12, 3], see also [13, 14, 9, 10, 11, 2]). It was asked in [5] for correspond...

. The multilevel method has emerged as one of the most effective methods for solving numerical and combinatorial problems. It has been used in multigrid, domain decomposition, geometric search structures, as well as optimization algorithms for problems such as partitioning and sparse-matrix ordering. This paper presents a systematic treatment of the fundamental elements of the multilevel method. We illustrate, using examples from several fields, the importance and effectiveness of coarsening, sampling, and smoothing (local optimization) in the application of the multilevel method. Key words. Algorithm-design paradigm, coarsening, combinatorial optimization, Delaunay triangulation, domain decomposition, eigenvalue problems, Gaussian elimination, geometric methods, graph partitioning, hierarchical methods, multigrid, multilevel methods, nested dissection, sampling, smoothing, spectral methods. AMS(MOS) subject classifications. Primary 1234, 5678, 9101112. 1. Introduction. The multilev...

The planarity theorems of MacLane and Whitney are extended to compact graph–like spaces. This generalizes recent results of Bruhn and Stein (MacLane’s Theorem for the Freudenthal compactification of a locally finite graph) and of Bruhn and Diestel (Whitney’s Theorem for an identification space obtained from a graph in which no two vertices are joined by infinitely many edge-disjoint paths). 1

We investigate the end spaces of infinite dual graphs. We show that there exists a natural homeomorphism between the end spaces of a graph and its dual, and that this homeomorphism maps thick ends to thick ends. Along the way, we prove that Tutte-connectivity is invariant under taking (infinite) duals.

Nash-Williams conjectured that a 4-connected infinite planar graph contains a spanning 2-way infinite path if, and only if, the deletion of any finite set of vertices results in at most two infinite components. In this paper, we prove the Nash-Williams conjecture for graphs with no dividing cycles and for graphs with infinitely many vertex disjoint dividing cycles. A cycle in an infinite plane graph is called dividing if both regions of the plane bounded by this cycle contain infinitely many vertices of the graph.