Drawing Planar Graphs Symmetrically, III: Oneconnected Planar Graphs

Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. This paper discusses symmetric drawings of oneconnected planar graphs. More specifically, we discuss planar (geometric) automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a planar drawing of G. Finding planar automorphisms is the first and most difficult step in constructing planar symmetric drawings of graphs. The problem of determining whether a given graph has a nontrivial geometric automorphism is NP-complete for general graphs. The two previous papers in this series have discussed the problem of drawing planar graphs with a maximum number of symmetries, for the restricted cases where the graph is triconnected and biconnected. This paper extends the previous results to cover planar graphs that are oneconnected. We present a linear time algorithm for drawing oneconnected planar graphs with a maximum number of symmetries.     

Drawing Planar Graphs Symmetrically, II: Biconnected Planar Graphs

Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals the structure in the graph. This paper discusses symmetric drawings of biconnected planar graphs. More specifically, we discuss geometric automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a drawing of G. Finding geometric automorphisms is the first and most difficult step in constructing symmetric drawings of graphs. The problem of determining whether a given graph has a non-trivial geometric automorphism is NP-complete for general graphs. In this paper we present a linear time algorithm for finding planar geometric automorphisms of biconnected planar graphs. A drawing algorithm is also discussed.     

基于遗传算法的平面图平面正交直线画图算法

提出了一种基于遗传算法的新的平面图平面正交直线画图算法,算法将平面图画图问题转化为约束优化问题,根据画图问题选定的美观准则构造约束函数,用遗传算法求解目标函数的最优解的近似值,从而得到平面图的平面正交直线画法。新算法的优点是方法简单,易于实现,画出的图形美观,算法稳定性好。实验结果表明,画图算法的最终结果不依赖于图的初始状态。     

On simultaneous straight-line grid embedding of a planar graph and its dual

Simultaneous representations of planar graphs and their duals normally require that the dual vertices to be placed inside their corresponding primal faces, and the edges of the dual graph to cross only their corresponding primal edges. Erten and Kobourov [C. Erten, S.G. Kobourov, Simultaneous embedding of a planar graph and its dual on the grid, Theory Computer Systems 38 (2005) 313-327] provided a linear time algorithm on simultaneous straight-line grid embedding of a 3-connected planar graph and its dual such that all the vertices are placed on grid points and each edge is drawn as one straight-line segment except for one which is drawn using two segments. Their drawing size is (2n−2)×(2n−2), where n is the total number of vertices in the graph and its dual. They raised an open question on whether there is a large class of planar graphs that allows this simultaneous straight-line grid embedding on a smaller grid. We answer this open question by giving a linear time simultaneous straight-line grid embedding algorithm for a 3-connected planar graph and its dual on a grid of size (n−1)×n.     

用遗传算法画无向图

本文提出了一个新的画一般无向图的遗传算法.以前的无向图画图算法将顶点数较多且无弦的圈画成了凹多边形,为了克服这一缺点,本文的遗传算法设计了全新的变异算子--单点邻域变异,并在适应度函数中增加用于产生对称画法的分量,可将这种图画成凸多边形.新算法的优点是方法简单,易于实现,画出的图形美观,其灵活之处在于准则的权重可以改变.实验结果表明,在相同条件下,本文算法画出的图形要比标准遗传算法画出的图形美观.     

Efficient Orthogonal Drawings of High Degree Graphs

Most of the work that appears in the two-dimensional orthogonal graph drawing literature deals with graphs whose maximum degree is four. In this paper we present an algorithm for orthogonal drawings of simple graphs with degree higher than four. Vertices are represented by rectangular boxes of perimeter less than twice the degree of the vertex. Our algorithm is based on creating groups / pairs of vertices of the graph. The orthogonal drawings produced by our algorithm have area at most (m-1) ( m / 2 +2) . Two important properties of our algorithm are that the drawings exhibit a small total number of bends (less than m ), and that there is at most one bend per edge. Received January 15, 1997; revised February 1, 1998.     

Small Area Drawings of Outerplanar Graphs

We show three linear-time algorithms for constructing planar straight-line grid drawings of outerplanar graphs. The first and the second algorithm are for balanced outerplanar graphs. Both require linear area. The drawings produced by the first algorithm are not outerplanar while those produced by the second algorithm are. On the other hand, the first algorithm constructs drawings with better angular resolution. The third algorithm constructs outerplanar drawings of general outerplanar graphs with O(n ^1.48) area. Further, we study the interplay between the area requirements of the drawings of an outerplanar graph and the area requirements of a special class of drawings of its dual tree. Work partially supported by MUR under Project MAINSTREAM Algorithms for Massive Information Structures and Data Streams.     

Drawing complete binary trees inside rectilinear polygons

Most of graph drawing algorithms draw graphs on unbounded planes. However, there are applications that require graphs to be drawn on the plane inside a given polygon. In this paper, a new algorithm for planar orthogonal drawing of complete binary trees inside rectilinear polygons is presented. Uniform distribution of nodes of graphs on drawing regions is one of the aesthetics criteria in graph drawing. The goal of this paper is to produce planar orthogonal drawings with a relatively uniform node distribution and few edge bends. The proposed algorithm can be considered as a generalization of the H-tree layout method for rectilinear polygons. A new linear time algorithm is also given for bisecting rectilinear polygons into two equi-area rectilinear sub-polygons.     

用多层次聚类法完成的大规模关系图的可视化

提出了一种新的大规模图形可视化技术.它可显示含有几万个接点和边的大规模关系图.为了完成对图形的抽象化。一个多层次的聚类图形从原始的大规模关系图中抽取了出来.这种抽取是建立在大规模关系图的内在结构基础上来完成的.一种递规封入式的几何划分算法被应用来完成对几何空间的优化,在具体的制图技术上,使用了一种用力导向布局算法和环形制图法相结合的新方法,从而完成了对显示空间的优化和美擘上的优化.同时也讨论了相关的人机交互技术,所采用的人机交互算法不仅能让使用者从上到下层次式地浏览整个聚类图形。同时也能提供多层次聚类图形的并行浏览.动画技术也同时被运用,以保护使用者的精神图不被打乱.     

Evaluating esthetics for user-sketched layouts of clustered graphs with known clustering information

This paper aims to empirically analyze the esthetics for user-sketched layouts of clustered graphs with known clustering information. In our experiments, given not only the adjacency list of a clustered graph but also its predefined clustering information, each participant was asked to manually sketch clustered graphs “nicely” from scratch on a tablet system using a stylus. Different from previous works, the main concern in this paper is on which graph drawing esthetics people favor when sketching their own drawings of clustered graphs with known clustering information. Another concern of this paper is on the esthetics of clustered graph layouts employed by participants which include not only characteristics and structures of the final graph layouts but also the behavior of user's sketching process (including layout creation and adjustment). By observing all layouts and drawing processes, the drawing strategies which participants applied and the drawing esthetics are analyzed. Results show that most participants were unsurprisingly able to draw graphs with clear presence of bridge edges and clustering cohesiveness; more importantly, to distinguish clusters within the restricted-size tablet screen during the drawing process, some of the participants were still able to make each cluster with fewer edge crossings, more symmetries, and more alignment of grid in a smaller drawing area where the cluster spreads. Our results support that to alleviate user's complex drawing tasks, aside from the grid-based editing function suggested by the previous work, graph drawing systems should also provide the clustering information if the structure of the graph to be drawn is known.     

An Algorithm for Straight-Line Drawing of Planar Graphs

We present a new algorithm for drawing planar graphs on the plane. It can be viewed as a generalization of the algorithm of Chrobak and Payne, which, in turn, is based on an algorithm by de Fraysseix, Pach, and Pollack. Our algorithm improves the previous ones in that it does not require a preliminary triangulation step; triangulation proves problematic in drawing graphs ``nicely,' as it has the tendency to ruin the structure of the input graph. The new algorithm retains the positive features of the previous algorithms: it embeds a biconnected graph of n vertices on a grid of size (2n-4) x (n-2) in linear time. We have implemented the algorithm as part of a software system for drawing graphs nicely. Received September 21, 1995; revised March 15, 1996.     

A Linear-Time Algorithm for Star-Shaped Drawings of Planar Graphs with the Minimum Number of Concave Corners

A star-shaped drawing of a graph is a straight-line drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. In this paper, we consider the problem of finding a star-shaped drawing of a biconnected planar graph with the minimum number of concave corners. We first show new structural properties of planar graphs to derive a lower bound on the number of concave corners. Based on the lower bound, we prove that the problem can be solved in linear time by presenting a linear-time algorithm for finding a best plane embedding of a biconnected planar graph with the minimum number of concave corners. This is in spite of the fact that a biconnected planar graph may have an exponential number of different plane embeddings.     

基于遗传算法的平面图画图算法

提出了一种基于遗传算法的新的平面图画图算法，算法将平面图画图问题转化为约束优化问题，用遗传算法求解目标函数的最优解的近似值，从而得到平面图的平面直线画法．新算法的优点是：方法简单，易于实现，画出的图形美观．实验结果表明：算法画出的图形要比文献[8]中的算法画出的图形美观，而其收敛性则要高于标准遗传算法．     

Mental map preserving graph drawing using simulated annealing

Visualizing graphs has been studied extensively in the community of graph drawing and information visualization over the years. In some applications, the user is required to interact with a graph by making slight changes to the underlying graph structure. To visualize graphs in such an interactive environment, it is desirable that the differences between the displays of the original and the modified graphs be kept minimal, allowing the user to comprehend the changes in the graph structure faster. As the mental map concept refers to the presentation of a person’s mind while exploring visual information, the better the mental map is preserved, the easier the structure change of a graph is understood. It is somewhat surprising that preserving the user’s mental map has largely been ignored in the graph drawing community in the past. We propose an effective mental-map-preserving graph drawing algorithm for straight-line drawings of general undirected graphs based on the simulated-annealing technique. Our experimental results and questionnaire analysis suggest this new approach to be promising.     

On-line Animated Visualization of Huge Graphs using a Modified Spring Algorithm

On-line graph drawing deals with huge graphs which are partially unknown. At any time, a tiny part of the graph is displayed on the screen. Examples include web graphs and graphs of links in distributed file systems. This paper discusses issues arising in the presentation of such graphs. The paper describes a system for dealing with web graphs using on-line graph drawing.     

Drawing graphs with nonuniform nodes using potential fields

Graphs with nonuniform nodes arise naturally in many real-world applications. Although graph drawing has been a very active research in the computer science community during the past decade, most of the graph drawing algorithms developed thus far have been designed for graphs whose nodes are represented as single points. As a result, it is of importance to develop drawing methods for graphs whose nodes are of different sizes and shapes, in order to meet the need of real-world applications. To this end, a potential field approach, coupled with an idea commonly found in force-directed methods, is proposed in this paper for drawing graphs with nonuniform nodes in 2-D and 3-D. In our framework, nonuniform nodes are uniformly charged, while edges are modelled by springs. Using certain techniques developed in the field of potential-based path planning, we are able to find analytically tractable procedures for computing the repulsive force and torque of a node in the potential field induced by the remaining nodes. The crucial feature of our approach is that the rotation of every nonuniform node and the multiple edges between two nonuniform nodes are taken into account. In comparison with the existing algorithms found in the literature, our experimental results suggest this new approach to be promising, as drawings of good quality for a variety of moderate-sized graphs in 2-D and 3-D can be produced reasonably efficiently. That is, our approach is suitable for moderate-sized interactive graphs or larger-sized static graphs. Furthermore, to illustrate the usefulness of our new drawing method for graphs with zero-sized nodes, we give an application to the visualization of hierarchical clustered graphs, for which our method offers a very efficient solution.     

一个新的无向图画图算法

将一般无向图的画图问题转化为函数优化问题,用遗传算法求目标函数的最优解的近似值,从而得到无向图自动画图算法的一个一般框架.新方法的特点是:不同的画图算法的框架都一样,所不同的只是反映无向图画图问题的美观标准的目标函数.其优点在于,算法统一、方法简单、容易实现、便于修改,并且易于并行化,可以直接用来画非连通图.     

Approximation Algorithms and Hardness Results for Packing Element-Disjoint Steiner Trees in Planar Graphs

We study the problem of packing element-disjoint Steiner trees in graphs. We are given a graph and a designated subset of terminal nodes, and the goal is to find a maximum cardinality set of element-disjoint trees such that each tree contains every terminal node. An element means a non-terminal node or an edge. (Thus, each non-terminal node and each edge must be in at most one of the trees.) We show that the problem is APX-hard when there are only three terminal nodes, thus answering an open question. Our main focus is on the special case when the graph is planar. We show that the problem of finding two element-disjoint Steiner trees in a planar graph is NP-hard. Similarly, the problem of finding two edge-disjoint Steiner trees in a planar graph is NP-hard. We design an algorithm for planar graphs that achieves an approximation guarantee close to 2. In fact, given a planar graph that is k element-connected on the terminals (k is an upper bound on the number of element-disjoint Steiner trees), the algorithm returns $\lfloor\frac{k}{2} \rfloor-1$ element-disjoint Steiner trees. Using this algorithm, we get an approximation algorithm for the edge-disjoint version of the problem on planar graphs that improves on the previous approximation guarantees. We also show that the natural LP relaxation of the planar problem has an integrality ratio approaching?2.     

一种用于Marching-Graph图形绘制的快速收敛布局算法

Marching-Graph是一种将图形隐喻技术和空间隐喻技术集成为一体的新的可视化方法.它为用户提供了高度可交互性地图,使用户可访问那些具有地理属性的信息的逻辑结构.它通过有效的人图交互和跨空间浏览为用户提供了一种可视分析和挖掘未知信息的机制,而不是将已知的信息呈现在地图上.然而,传统的力导向布局算法在达到力量均衡配置方面非常慢.为使一个图形布局收敛,它们通常需花费几十秒的时间.因此。当用户快速行进于地理区间时,那些力导向布局算法就不能满足快速绘制一系列图形的要求.提出了一种快速收敛布局方法,当用户在Marching-Graph中通过力导向布局逐步探究一系列图形时,它可以加速交互时间.通过结合辐射树绘图技术和力导向图形绘制方法来取得能量最小化的快速收敛.     

Fast 3-coloring Triangle-Free Planar Graphs

Although deciding whether the vertices of a planar graph can be colored with three colors is NP-hard, the widely known Grötzsch’s theorem states that every triangle-free planar graph is 3-colorable. We show the first o(n ²) algorithm for 3-coloring vertices of triangle-free planar graphs. The time complexity of the algorithm is $\mathcal{O}(n\log n)Although deciding whether the vertices of a planar graph can be colored with three colors is NP-hard, the widely known Gr?tzsch’s theorem states that every triangle-free planar graph is 3-colorable. We show the first o(n ²) algorithm for 3-coloring vertices of triangle-free planar graphs. The time complexity of the algorithm is O(nlogn)\mathcal{O}(n\log n) .