Graph edit distance: Restrictions to be a metric

In the presentation of the graph edit distance in 1983 and other newer bibliography, authors state that it is necessary to apply the distance restrictions (non-negativity, identity of indiscernible elements, symmetry and triangle inequality) to each of the edit functions (insertion, deletion and substitution of nodes and edges) involved in the process of computing the graph edit distance to make the graph edit distance a true distance. Moreover, graph edit distance algorithms presented in the last three decades have been based on mapping the edit path that transforms a graph into the other one into a bijection of the graphs in which some null nodes have been added. In this paper, we show that the triangle inequality does not need to be imposed in each edit function if the graph edit distance is defined through an edit path; however, it is necessary if it is defined as a graph bijection. This is an important finding since the triangle inequality is the only restriction that relates different edit functions and concerns the process of tuning the edit functions given a specific application. Hence, on one hand, it would encourage research to define new algorithms based on the edit path instead of the graph bijection and, on the other hand, use edit functions without the restriction, for instance, that the sum of the costs of insertion and deletion of a pair of nodes has to be larger or equal than the cost of substituting them, which could increase the recognition ratio of a concrete application. (C) 2019 Elsevier Ltd. All rights reserved.