Transforming a random graph drawing into a Lombardi drawing

Abstract: 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. A Lombardi drawing of a graph is a drawing where the edges are drawn as circular arcs (straight edges are considered circular arcs of infinite radius) 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 $2\pi/d$ where d is the degree of that specific vertex. 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, giving a proper visualization of the graph.