Towards shortest longest edges in orthogonal graph drawing

Inspired by a challenge during Graph Drawing 2010 "Find an orthogonal drawing whose longest edge is as short as possible", we investigate techniques to incorporate this goal into the "standard" topology-shape-metrics approach at moderate extra computational complexity. Experiments indicate that this project is worth pursuing

Steinitz Theorems for Orthogonal Polyhedra

We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we find graph-theoretic characterizations of three classes of simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex, and arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz polyhedra are exactly the bipartite cubic polyhedral graphs, and every bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of a corner polyhedron. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure

An automatic generation of metro-like maps to display flight routes for air traffic controllers: structure and color optimization

International audienceAircraft must follow strict Air Traffic Control (ATC) rules. One of these rules is that aircraft have to fly over pre-defined Flight Routes (FR). Current ATC visualizations do not display FRs because they are numerous and run into each other, and thus spoil the visualization. The schematic views for metro maps are used to maximize the transmission of relevant information (lines, metro stops) of network visualization. In this paper, we will focus on two different issues. First, we show how we transposed mathematical constraints used to produce metro maps into the specific field of ATC. The view produced is a context compatible, 2D picture of a schematic maps view for Air Traffic Control. Second, we propose to investigate the generation and placement of colors to be assigned to lines of the network. The first step is to find as many colors as lines of the network. These colors must be perceptually as distinct as possible, and available in the vocabulary of colors. The second step is to solve the NP-complete problem of the optimal assignment of these colors so that close lines have the most perceptively distant color. Finally, we assess the map produced through experimentation to validate its quality