A sequence of nonnegative integers S=(s1, s2, , sn) is graphical if there is a graph with vertices v1,v2, ,vn such that deg(vi)=si for each i=1, 2, , n. The graphical degree sequence problem is： Given a sequence of nonnegative integers, determine whether it is graphical or not. In this paper, we consider several variations of the graphical degree sequence problem and give efficient algorithms.

The label placement problem is one of the most important problems in geographic information systems, cartography, graph drawing and graphical interface design. In this paper, we consider the problem of labeling points and curves in maps drawn from digital data. In digital maps, a curve is represented as a set of points and consists of many small segments. The label for each curve must be placed alongside the corresponding curve. We define a continuous labeling space for points and curves, and present an algorithm using this space for positioning labels. Computational results for subway and JR train maps in Tokyo are presented.

This paper reports computational results of a new approach of analyzing network reliability against probabilistic link failures. This problem is hard to solve exactly when it is large-scale, which is shown from complexity theory, but the approach enables us to analyze networks of moderate size, as demonstrated by our experimental results. Furthermore, this approach yields a polynomial-time algorithm for complete graphs, whose reliability provides a natural upper bound for simple networks, and also leads to an efficient algorithm for computing the dominant part of the reliability function when the failure probability is sufficiently small. Computational results for these cases are also reported. This approach thus establishes a fundamental technology of analyzing network reliability in practice.

This paper formulates problems of fitting two corresponding sets of points by translation, rotation and scaling, and proposes efficient algorithms for the fitting. The algorithms are based on the theory of lower envelopes, or Davenport-Schinzel sequences, and linearization techniques in computational geometry, and are related to dynamic furthest Voronoi diagrams.

The label placement problem is one of the most important problems in geographic information systems, cartography, graph drawing, and graphical interface design. In this paper, we considered the label size maximization problem for points with axes-parallel rectangular labels that correspond to character strings and have different widths based on the number of characters. We propose an algorithm for computing the optimum size for the label size maximization problem in the 2-position model and a polynomial time algorithm for the problem in the 4-position model. Our algorithm cannot obtain the maximum value in the 4-position model because the label size maximization problem in the 4-position model is NP-hard. However, our algorithm is efficient in practice, as shown by computational experiments. Further, computational results for JR trains, subways and major private railroads in Tokyo are presented.

Triangulations have been one of main research topics in computational geometry and have many applications in computer graphics, finite element methods, mesh generation, etc. This paper surveys properties of triangulations in the two- or higher-dimensional spaces. For triangulations of the planar point set, we have a good triangulation, called the Delaunay triangulation, which satisfies several optimality criteria. Based on Delaunay triangulations, many properties of planar triangulations can be shown, and a graph structure can be constructed for all planar triangulations. On the other hand, triangulations in higher dimensions are much more complicated than in planar cases. However, there does exist a subclass of triangulations, called regular triangulations, with nice structure, which is also touched upon.

The Tokyo subway is one of the most complex subway networks in the world and it is difficult to compute a visually readable metro map using existing layout methods. In this paper, we present a new method that can generate complex metro maps such as the Tokyo subway network. Our method consists of two phases. The first phase generates rough metro maps. It decomposes the metro networks into smaller subgraphs and partially generates rough metro maps. In the second phase, we use a local search technique to improve the aesthetic quality of the rough metro maps. The experimental results including the Tokyo metro map are shown.