We present a real-time algorithm for computing the Voronoi diagram of planar freeform piecewise-spiral curves. The efficiency and robustness of our algorithm is based on a simple topological structure of Voronoi cells for spirals. Using a Möbius transformation, we provide an efficient search for maximal disks. The correct topology of Voronoi diagram is computed by sampling maximal disks systema...

We investigate the existence of deterministic uniform self-stabilizing algorithms (DUSSAs) for a number or problems on connected undirected graphs. This investigation is carried out under three models of parallelism, namely central daemon, restricted parallelism, and maximal paral-lelism. We observe that for several problems including 2-coloring odd-degree complete bipartite graphs, 2-coloring ...

A graph is NIC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex. NIC-planarity generalizes IC-planarity, which allows a vertex to be incident to at most one crossing edge, and specializes 1-planarity, which only requires at most one crossing per edge. We characterize embeddings of maximal ...

Based on a new version of Hopcroft and Tarjan’s planarity testing algorithm, we develop an O (mlogn)-time algorithm to find a maximal planar subgraph.

Abstmct. Wagner’s theorem (any two maximal plane graphs having p vertices are equivalent under diagonal transformations) is extended to maximal torus graphs, graphs embedded in the torus with a maximal set of edges present. Thus any maximal torus graph havingp vertices may be diagonally transformed into any other maximal torus p;aph having p vertices. As with Wag mr’s theorem, a normal form rep...

Three types of geometric structure—grid triangulations, rectangular subdivisions, and orthogonal polyhedra— can each be described combinatorially by a regular labeling : an assignment of colors and orientations to the edges of an associated maximal or near-maximal planar graph. We briefly survey the connections and analogies between these three kinds of labelings, and their uses in designing ef...

Given a graph G, the NP-hard Maximum Planar Subgraph problem (MPS) asks for a planar subgraph of G with the maximum number of edges. There are several heuristic, approximative, and exact algorithms to tackle the problem, but—to the best of our knowledge— they have never been compared competitively in practice. We report on an exploratory study on the relative merits of the diverse approaches, f...

Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of G-H can be added to H without destroying planarity. Polynomial algorithms have been obtained by Jakayumar, Thulasiraman and Swamy [6] and Wu [9]. O(mlogn) algorithms were previously given by Di Battista and Tamassia [3] and Cai, Han and Tarjan [2]. A recent O(m α α (n))...

The (∆, D) (degree/diameter) problem consists of finding the largest possible number of vertices n among all the graphs with maximum degree ∆ and diameter D. We consider the (∆, D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We prove that for the (∆, 2) problem, the number of vertices is n = ∆ + 2; and for the (∆, 3) problem, n...