We give an overview of the LEDA platform for combinatorial and geometric computing and an account of its development. We discuss our motivation for building LEDA and to what extent we have reached our goals. We also discuss some recent theoretical developments. This paper contains no new technical material. It is intended as a guide to existing publications about the system. We refer the reader also to our web-pages for more information.

This is a survey on graph visualization and navigation techniques, as used in information visualization. Graphs appear in numerous applications such as web browsing, state--transition diagrams, and data structures. The ability to visualize and to navigate in these potentially large, abstract graphs is often a crucial part of an application. Information visualization has specific requirements, which means that this survey approaches the results of traditional graph drawing from a different perspective. Index Terms---Information visualization, graph visualization, graph drawing, navigation, focus+context, fish--eye, clustering. 1

Petersen's theorem is a classic result in matching theory from 1891, stating that every 3-regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n^3/2) time for 3-regular graphs. We have developed an O(n log^4 n)-time algorithm for perfect matching in a 3-regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)-time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation.

Given an undirected graph, the planarity testing problem is to determine whether the graph can be drawn in the plane without any crossing edges. Linear time planarity testing algorithms have previously been designed by Hopcroft and Tarjan, and by Booth and Lueker. However, their approaches are quite involved. Several other approaches have also been developed for simplifying the planariy test. In this paper, we developed a very simple linear time testing algorithm based only on a depth-first search tree. When the given graph is not planar, our algorithm immediately produces explicit Kuratowski's subgraphs. A new data structure, PC-trees, is introduced, which can be viewed as abstract subembeddings of actual planar embeddings. A graph-reduction technique is adopted so that the embeddings for the planar biconnected components constructed at each iteration never have to be changed. The recognition and embedding are actually done simultaneously in our algorithm 1 . The implementation of o...

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Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...

: We present a new algorithm for drawing planar graphs on the plane. It can be viewed as a generalization of the algorithm of Chrobak and Payne, which in turn, is based on an algorithm by de Fraysseix, Pach and Pollack. Our algorithm improves the previous ones in that it does not require a preliminary triangulation step; triangulation proves problematic in drawing graphs "nicely", as it has the tendency to ruin the structure of the input graph. The new algorithm retains the positive features of the previous algorithms: It embeds a biconnected graph of n vertices on a grid of size (2n \Gamma 4) \Theta (n \Gamma 2) in linear time. We have implemented the algorithm as part of a software system for drawing graphs nicely. 1 Introduction In this paper we describe a new drawing algorithm for planar graphs. The algorithm is a central component of a software system we have developed for drawing graphs "nicely" [HS94], and was especially designed for that purpose. (The other main component of ...

Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e in which all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a planar combinatorial embedding of a planar graph G in which the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NP-hard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQR-trees, which is able to find a crossing minimum solution.

We show that the edges of a graph with maximum edge density d can always be oriented such that each vertex has in-degree at most d. Hence, for arbitrary graphs, edges can always be assigned to incident vertices as uniformly as possible. For example, in-degree 3 is achieved for planar graphs. This immediately gives a space-optimal data structure that answers edge membership queries in a maximum edge density-d graph in O(log d) time. Keywords Graph orientation, edge density, Hall condition, balanced adjacency lists, edge membership queries 1 The Theorem Let G be an undirected graph with n vertices and m edges. The parameter ffi(G) = m n is commonly called the edge density of G. The maximum (edge) density is the smallest integer d such that the edge density of any subgraph of G does not exceed d. More precisely, d = dmaxfffi(G 0 ) j G 0 is a subgraph of Gge. For example, d 1 for trees, d 3 for planar graphs, d = d 1 2 log 2 ne for hypercubes [GG,AH], and d d 1 2 (c \Gamma...

We report on the use of program checking in the LEDA library of efficient data types and algorithms.