We study the algorithmic problem of computing drawings of graphs in which $(i)$ each vertex is a disk with fixed radius $\rho$, $(ii)$ each edge is a straight-line segment connecting the centers of the two disks representing its end-vertices, $(iii)$ no two disks intersect, and $(iv)$ the distance between an edge segment and the center of a non-incident disk, called \emph{edge-vertex resolution}, is at least $\rho$. We call such drawings \emph{disk-link drawings}. In this paper we focus on the case of constant edge-vertex resolution, namely $\rho=\frac{1}{2}$ (i.e., disks of unit diameter). We prove that star graphs, which trivially admit straight-line drawings in linear area, require quadratic area in any such disk-link drawing. On the positive side, we present constructive techniques that yield improved upper bounds for the area requirements of disk-link drawings for several (planar and nonplanar) graph classes, including bounded bandwidth, complete, and planar graphs. In particular, the presented bounds for complete and planar graphs are asymptotically tight.
翻译:我们研究计算图纸图纸的算法问题,其中每个顶端至少有$(一)美元是一个固定半径为$\rho$的磁盘,每个边缘有美元(二)是一个直线段连接两个磁盘中心以代表其最终垂直的圆盘中心的部分,一个(三)美元没有两个磁盘交叉的圆盘,另一个(四)美元是边缘部分与非事件磁盘中心之间的距离,称为\emph{ge-vertex分辨率},至少是$(rho)美元。在正面方面,我们提出建设性技术,为磁盘链图的面积要求带来改进,即:$\rho{frac{%1}2}(即,单位直径的磁盘)。我们证明,在线性区域中微不足道地接受直线线线图的恒星图需要四面区域。在正面方面,我们提出建设性技术,使磁盘链接图纸绘制区域要求的上边框得到改进,包括一些磁盘和特定平面平面平面图的平面图,包括若干平面平面图和非平面图。