We consider the problem of orienting the edges of a planar graph in such a way that the out-degree of each vertex is minimized. If, for each vertex v, the out-degree is at most d, then we say that such an orientation is d-bounded. We prove the following results: ffl Each planar graph has a 5-bounded acyclic orientation, which can be constructed in linear time. ffl Each planar graph has a 3-bounded orientation, which can be constructed in linear time. ffl A 6-bounded acyclic orientation, and a 3-bounded orientation, of each planar graph can each be constructed in parallel time O(log n log n) on an EREW PRAM, using O(n= log n log n) processors. As an application of these results, we present a data structure such that each entry in the adjacency matrix of a planar graph can be looked up in constant time. The data structure uses linear storage, and can be constructed in linear time. Department of Mathematics and Computer Science, University of California, Riverside, CA 92521. On...

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide—to the best of our knowledge—the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.

Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in software engineering, project management, and visual languages. In this paper we investigate upward planarity testing of single-source digraphs; we provide a new combinatorial characterization of upward planarity and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a single-source digraph with n vertices is upward planar in O(n) sequential time, and in O(log n) time on a CRCW PRAM with n log log n / log n processors, using O(n) space. The algorithm also constructs an upward planar drawing if the test is successful. The previously known best result is an O(n2)-time algorithm by Hutton and Lubiw [Proc. 2nd ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1991, pp. 203–211]. No efficient parallel algorithms for upward planarity testing were previously known.

We consider a modified notion of planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planarity requires). Such adjacencies define a map graph. We give an NP characterization for such graphs, and an O(n³)-time recognition algorithm for a restricted version: given a graph, decide whether it is realized by adjacencies in a map without holes, in which at most four nations meet at any point.

We show that for every fixed��there is a quadratic time algorithm that decides whether a given graph has crossing number at most�and, if this is the case, computes a drawing of the graph into the plane with at most�crossings. 1.

We consider several variations of the simultaneous embedding problem for planar graphs. We begin with a simple proof that not all pairs of planar graphs have simultaneous geometric embedding. However, using bends, pairs of planar graphs can be simultaneously embedded on the O(n 2) × O(n 2) grid, with at most three bends per edge, where n is the number of vertices. The O(n) time algorithm guarantees that two corresponding vertices in the graphs are mapped to the same location in the final drawing and that both the drawings are crossing-free. The special case when both input graphs are trees has several applications, such as contour tree simplification and evolutionary biology. We show that if both the input graphs are are trees, only one bend per edge is required. The O(n) time algorithm guarantees that both drawings are crossings-free, corresponding tree vertices are mapped to the same locations, and all vertices (and bends) are on the O(n 2) × O(n 2) grid (O(n 3) × O(n 3) grid). For the special case when one of the graphs is a tree and the other is a path we can find simultaneous embedding with fixed-edges. That is, we can guarantee that corresponding vertices are mapped to the same locations and that corresponding edges are drawn the same way. We describe an O(n) time algorithm for simultaneous embedding with fixededges for tree-path pairs with at most one bend per tree-edge and no bends along path edges, such that all vertices (and bends) are on the O(n) × O(n 2) grid, (O(n 2) × O(n 3) grid).

Key words: Graph, depth-first search, strongly connected component, biconnected component, stack.

For any fixed parameter t _> 1, a t-spanner of a graph G is a spanning subgraph in which the distance between every pair of vertices is at most t times their distance in G. A minimum t-spanner is a t-spanner with minimum total edge weight or, in unweighted graphs, minimum number of edges. In this paper, we prove the AlP-hardness of finding minimum t-spanners for planar weighted graphs and digraphs if t _> 3, and for planar unweighted graphs and digraphs if t _> 5. We thus extend results on that problem to the interesting case where the instances are known to be planar. We also introduce the related problem of finding minimum planar t-spanners and establish its Alp-hardness for similar fixed values of t.

. We present a linear time algorithm that constructs a planar polyline grid drawing of any plane graph with n vertices and maximum degree d on a (2n \Gamma 5) \Theta ( 3 2 n \Gamma 7 2 ) grid with at most 5n \Gamma 15 bends and minimum angle ? 2 d . In the constructed drawings, every edge has at most three bends and length O(n). To our best knowledge, this algorithm achieves the best simultaneous bounds concerning the grid size, angular resolution, and number of bends for planar grid drawings of high-degree planar graphs. Besides the nice theoretical features, the practical drawings are aesthetically very pleasing. An implementation of our algorithm is available with the AGD-Library (Algorithms for Graph Drawing) [2, 1]. Our algorithm is based on ideas by Kant for polyline grid drawings for triconnected plane graphs [23]. In particular, our algorithm significantly improves upon his bounds on the angular resolution and the grid size for non-triconnected plane graphs....

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.