Orthogonal drawings, i.e., embeddings of graphs into grids, are a classic topic in Graph Drawing. Often the goal is to find a drawing that minimizes the number of bends on the edges. A key ingredient for bend minimization algorithms is the existence of an orthogonal representation that describes such drawings combinatorially by only listing the angles between the edges around each vertex and the directions of bends on the edges, but neglecting any kind of geometric information such as vertex coordinates or edge lengths. We generalize this idea to ortho-radial representations of ortho-radial drawings, which are embeddings into an ortho-radial grid, whose gridlines are concentric circles around the origin and straight-line spokes emanating from the origin but excluding the origin itself. Unlike the orthogonal case, there exist ortho-radial representations that do not admit a corresponding drawing, for example so-called strictly monotone cycles. An ortho-radial drawing is called valid if it does not contain a strictly monotone cycle. Our first result is that an ortho-radial representation admits a corresponding drawing if and only if it is valid. Previously such a characterization was only known for ortho-radial drawings of paths, cycles, and theta graphs, and in the special case of rectangular drawings of cubic graphs, where the contour of each face is required to be a rectangle. Further, we give a quadratic-time algorithm that tests for an ortho-radial representation whether it is valid, and we show how to draw a valid ortho-radial representation in the same running time. Altogether, this reduces the problem of computing a minimum-bend ortho-radial drawing to the task of computing a valid ortho-radial representation with the minimum number of bends, and hence establishes an ortho-radial analogue of the topology-shape-metrics framework for planar orthogonal drawings by Tamassia.
翻译:矩形绘图, 即: 将图形嵌入矩形坐标或边缘长度, 是一个典型的话题 。 我们的目标通常是找到一个能够最大限度地减少边缘弯曲的绘图。 弯曲最小化算法的关键元素是存在一个正方形图解, 仅通过列出每个顶端边缘和边缘弯面方向之间的角, 却忽略了任何种类的几何信息, 如角形坐标或边缘长度 。 我们把这个想法推广到正方形轨道图解的正方形图解 。 正在将正方形轨道图解嵌入正方形- 正方形图解的正向正方形图解, 其正方形图解的正方形和直线图解介的正方形图 。 与正方形图解的正方形图解不同, 以直立式框架或直方形图解的直方形图解的直径直径直径直径直径直径直径直径直的直径直径方形图解, 或直径直径直径直径直径直径直立的直径直径直径方形图解的图解的直径或直线形图解的直径直线形图解, 或直线形图解的直路径直地或直的直的直的直的直线形图解的直路路路路路路路路路路路路或直路路或直路路路路路路路路路路路路路路, 。