8 research outputs found
グラフアルゴリズムの効率化と評価に関する研究
平成13年度-平成14年度科学研究費補助金(基盤研究(C)(2))研究成果報告書,課題番号.1367038
Upward planar drawings with two slopes
In an upward planar 2-slope drawing of a digraph, edges are drawn as
straight-line segments in the upward direction without crossings using only two
different slopes. We investigate whether a given upward planar digraph admits
such a drawing and, if so, how to construct it. For the fixed embedding
scenario, we give a simple characterisation and a linear-time construction by
adopting algorithms from orthogonal drawings. For the variable embedding
scenario, we describe a linear-time algorithm for single-source digraphs, a
quartic-time algorithm for series-parallel digraphs, and a fixed-parameter
tractable algorithm for general digraphs. For the latter two classes, we make
use of SPQR-trees and the notion of upward spirality. As an application of this
drawing style, we show how to draw an upward planar phylogenetic network with
two slopes such that all leaves lie on a horizontal line
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
Drawing planar graphs with prescribed face areas
This thesis deals with planar drawings of planar graphs such that each interior face has
a prescribed area.
Our work is divided into two main sections. The rst one deals with straight-line drawings
and the second one with orthogonal drawings.
For straight-line drawings, it was known that such drawings exist for all planar graphs
with maximum degree 3. We show here that such drawings exist for all planar partial 3-trees,
i.e., subgraphs of a triangulated planar graph obtained by repeatedly inserting a vertex in
one triangle and connecting it to all vertices of the triangle. Moreover, vertices have rational
coordinates if the face areas are rational, and we can bound the resolution.
For orthogonal drawings, we give an algorithm to draw triconnected planar graphs with
maximum degree 3. This algorithm produces a drawing with at most 8 bends per face and
4 bends per edge, which improves the previous known result of 34 bends per face. Both
vertices and bends have rational coordinates if the face areas are rational
Orthogonal Drawings of Plane Graphs without Bends
In an orthogonal drawing of a plane graph each vertex is drawn as a point and each edge is drawn as a sequence of vertical and horizontal line segments. A bend is a point at which the drawing of an edge changes its direction. Every plane graph of the maximum degree at most four has an orthogonal drawing, but may need bends. A simple necessary and sufficient condition has not been known for a plane graph to have an orthogonal drawing without bends. In this paper we obtain a necessary and sufficient condition for a plane graph G of the maximum degree three to have an orthogonal drawing without bends. We also give a linear-time algorithm to find such a drawing of G if it exists