Abstract We give an overview of multilevel methods, such as V-cycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasi-uniform meshes and extend them to graded meshes and completely unstructured grids. We first discuss the classical multigrid theory on the basis of the method of subspace correction of Xu and a key identity of Xu and Zikatanov. We next extend the classical multilevel methods in H(grad) to graded bisection grids upon employing the decomposition of bisection grids of Chen, Nochetto, and Xu. We finally discuss a class of multilevel preconditioners developed by Hiptmair and Xu for problems discretized on unstructured grids and extend them to H(curl) and H(div) systems over graded bisection grids. 1

We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equiangularly spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs. Submitted:

ABSTRACT. This is the documentation of the local mesh refinement using newest bisection or longest bisection in MATLAB. The new feature of our implementation is the edge marking strategy to ensure the conformity. The short implementation is helpful for the teaching of adaptive finite element methods and programming in more advanced languages. 1.

A k-factor of graph G is defined as a k-regular spanning subgraph of G. For instance, a 2-factor of G is a set of cycles that span G. 2-factors have multiple applications in Graph Theory, Computer Graphics, and Computational Geometry [5, 4, 6, 11]. We define a simple 2-factor as a 2-factor without degenerate cycles. In general, simple k-factors are defined as k-regular spanning subgraphs where no edge is used more than once. We propose a new algorithm for computing simple k-factors for all values of k ≥ 2. 1

Abstract. We design and analyze optimal additive and multiplicative multilevel methods for solving H 1 problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs to be performed only for three vertices- the new vertex and its two parent vertices. We show that our methods lead to optimal complexity for any dimensions and polynomial degree. The theory hinges on a new decomposition of bisection grids in any dimension, which is of independent interest and yields a corresponding decomposition of spaces. We use the latter to bridge the gap between graded and quasi-uniform grids, for which the multilevel theory is well-established.

We describe a linear-time algorithm for 4-coloring planar graphs. We indeed give an O(V + E + |χ | + 1)-time algorithm to C-color V-vertex E-edge graphs embeddable on a 2-manifold M of Euler characteristic χ where C(M) is given by Heawood’s (minimax optimal) formula. Also we show how, in O(V + E) time, to find the exact chromatic number of a maximal planar graph (one with E = 3V − 6) and a coloring achieving it. Finally, there is a linear-time algorithm to 5-color a graph embedded on any fixed surface M except that an M-dependent constant number of vertices are left uncolored. All the algorithms are simple and practical and run on a deterministic pointer machine, except for planar graph 4-coloring which involves enormous constant factors and requires an integer RAM with a random number generator. All of the algorithms mentioned so far are in the ultra-parallelizable deterministic computational complexity class “NC. ” We also have more practical planar 4-coloring algorithms that can run on pointer machines in O(V log V) randomized time and O(V) space, and a very simple deterministic O(V)-time coloring algorithm for planar graphs which conjecturally uses 4 colors.

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Abstract. Many problems in computational geometry are not stated in graphtheoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm depends on the special properties of the graph constructed in this way. We survey the art gallery problem, partition into rectangles, minimum-diameter clustering, rectilinear cartogram construction, mesh stripification, angle optimization in tilings, and metric embedding from this perspective. 1