We make several observations on the implementation of Edmonds’ blossom algorithm for solving minimum-weight perfectmatching problems and we present computational results for geometric problem instances ranging in size from 1,000 nodes up to 5,000,000 nodes. A key feature in our implementation is the use of multiple search trees with an individual dual-change � for each tree. As a benchmark of the algorithm’s performance, solving a 100,000-node geometric instance on a 200 Mhz Pentium-Pro computer takes approximately 3 minutes.

Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have sphericity at most 3, is NP-hard. We show how this reduction can be extended to 3 dimensions, thereby showing that unit sphere graph recognition, or determining if a graph has sphericity 3 or less, is also NP-hard. We conjecture that K-sphericity is NP-hard for all fixed K greater than 1. 1 Introduction A unit disk graph is the intersection graph of a set of unit diameter closed disks in the plane. That is, each vertex corresponds to a disk in the plane, and two vertices are adjacent in the graph if the corresponding disks intersect. The set of disks is said to realize the graph. Of course, the unit of distance is not critical, since the disks realize the same graph even if the coordina...

Graph theory owes many powerful ideas and constructions to geometry. Several well-known families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another source of graphs are geometric configurations where the relation of incidence determines the adjacency in the graph. Interesting graphs possess some inner structure which allows them to be described by labeling smaller graphs. The notion of covering graphs is explored.

Abstract. We propose the DISCO algorithm for graph realization in Rd, given sparse and noisy short-range inter-vertex distances as inputs. Our divide-and-conquer algorithm works as follows. When a group has a sufficiently small number of vertices, the basis step is to form a graph realization by solving a semidefinite program. The recursive step is to break a large group of vertices into two smaller groups with overlapping vertices. These two groups are solved recursively, and the sub-configurations are stitched together, using the overlapping atoms, to form a configurations for the larger group. At intermediate stages, the configurations are improved by gradient descent refinement. The algorithm is applied to the problem of determining protein molecule structure. Tests are performed on molecules taken from the Protein Data Bank database. Given 20–30 % of the interatom distances less than 6˚A that are corrupted by a high level of noise, DISCO is able to reliably and efficiently reconstruct the conformation of large molecules. In particular, given 30 % of distances with 20 % multiplicative noise, a 13000-atom conformation problem is solved within an hour with an RMSD of 1.6˚A. 1. Introduction. A

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b-dimensional cube is a Cartesian product I1 × I2 × · · · × Ib, where each Ii is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G) is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b-dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line- i.e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m + 1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ⌈log 2 ψ(G)⌉. In this paper, we show that, for an interval graph G ⌈log 2 ψ(G) ⌉ ≤ cub(G) ≤ ⌈log 2 ψ(G) ⌉ + 2. It is not clear whether the upper bound of ⌈log 2 ψ(G) ⌉ + 2 is tight: Till now we are unable to find any interval graph with cub(G)> ⌈log 2 ψ(G)⌉. We also show that, for an interval graph G, cub(G) ≤ ⌈log 2 α⌉, where α is the independence number of G. Therefore, in the special case of ψ(G) = α, cub(G) is exactly ⌈log 2 α⌉. The concept of cubicity can be generalized by considering boxes instead of cubes. A b-dimensional box is a Cartesian product I1 ×I2 × · · ·×Ib, where each Ii is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k-dimensional boxes. It is clear that box(G) ≤ cub(G). From the above result, it follows that for any graph G, cub(G) ≤ box(G) ⌈log 2 α⌉.