Nice drawings of graphs are computationally hard

Abstract

How to draw a graph? And more importantly, how to draw it nicely? As a formal approach to this problem we propose graph embeddings. A graph embedding is a mapping from a guest graph into a host graph. Graph embeddings are very rich in their descriptive capabilities. These should suffice to capture all instances from real applications in an appropriate way. Graph embeddings offer various parameters for optimizations, which are used to describe aestetics in a formal and uniform way. Thus, we measure the niceness of a drawing by the values of its aestetic parameters, such as area, width, expansion, maximal and total edge length, or non-planarity. However, in this general framework and from an algorithmic point of view optimal embeddings or equivalently nice drawings of graphs are intractable. In general, they are NP-complete, which means that one must pay for nice drawings with a high computational effort. This fact holds even for trees. To the contrary, there are drawings of trees which satisfy the upper and lower bounds up to some constant factor and are computable in polynomial time.