The edit distance in graphs: methods, results and generalizations

The edit distance is a very simple and natural metric on the space of graphs. In the edit distance problem, we fix a hereditary property of graphs and compute the asymptotically largest edit distance of a graph from the property. This quantity is very difficult to compute directly but in many cases, it can be derived as the maximum of the edit distance function. Szemerédi’s regularity lemma, stronglyregular graphs, constructions related to the Zarankiewicz problem – all these play a role in the computing of edit distance functions. The most powerful tool is derived from symmetrization, which we use to optimize quadratic programs that define the edit distance function. In this paper, we describe some of the most common tools used for computing the edit distance function, summarize the major current results, outline generalizations to other combinatorial structures, and pose some open problems.