Transforming a random graph drawing into a Lombardi drawing

The visualization of any graph plays important role in various aspects, such as graph drawing software. Complex systems (like large databases or networks) that have a graph structure should be properly visualized in order to avoid obfuscation. One way to provide an aesthetic improvement to a graph visualization is to apply a force-directed drawing algorithm to it. This method, that emerged in the 60’s views graphs as spring systems that exert forces (repulsive or attractive) to the nodes. The first force-directed drawing algorithm of Tutte [1] was aiming at providing a barycentric representation of the graph and it is suitable for getting a straight-line drawing of 3-connected planar graphs without crossings. Each vertex is gradually placed at the berycenter of its neighbour vertices. This method used no spring systems, though it can be easily understood that springs can be applied to the system getting the same result as that of the barycentric method. Unfortunately, Tutte’s method usually gives bad angular resolutions. Many other force-directed methods have appeared since then. Eades’ method [2] creates 2D layouts using spring systems and it is suitable for graphs with less than 30 nodes. Attractive forces are applied to adjacent vertices only and repulsive to all pairs of vertices (so we can view the edges as springs with zero rest length and the nodes as electrically charged identical particles). Of course, the exact type of the forces exerted is a very important aesthetic criterion. Eades sets logarithmic attractive forces and inverse square root repulsive forces. Fruchterman and Reingold [3] apply the notion of simulated annealing, in order to provide an even better 2D layout of less cumulative spring energy, compared to that of Eades. The main disadvantage is, again, that the algorithm is limited to graphs having less than 40 vertices. Hadany and Harel [4] provide a force-directed layout for much larger graphs of over 1000 vertices. In order to obtain that, they follow a multi-scale approach where they initially consider an abstraction of the graph, which is iteratively augmented by the graph details. Gajer, Goodrich and Kobourov [5] use a coarsening strategy which avoids quadratic space and time complexities. They use a vertex filtration which enables the algorithm to restrict the number of vertices relocated. They, also, introduce the idea of realizing the graph in high-dimensional Euclidean space. An excellent survey on various force-directed methods can be found in [6]. A Lombardi drawing of a graph is a drawing where the edges are drawn as circular arcs (straight edges are considered degenerate circular ones) with perfect angular resolution. This means, that consecutive edges around a vertex are equally spaced around it. In other words, each angle between the tangents of two consecutive edges is equal to 2π/d where d is the degree of that specific vertex. This kind of drawing took its name from the American artist Mark Lombardi, who mainly used this technique in his paintings. The requirement of using circular edges in graphs when we want to provide perfect angular resolution is necessary, since even cycle graphs cannot be drawn with straight edges when perfect angular resolution is needed. In this survey, we provide an algorithm that takes as input a random drawing of a graph and provides its Lombardi drawing. 1 every angle in this survey will be expressed in rads