Multicolor and directed edit distance

TL;DR

This paper generalizes the concept of graph editing to multicolored and directed graphs, analyzing the minimum number of edge recolorings needed to achieve hereditary properties using advanced combinatorial and probabilistic methods.

Contribution

It introduces a framework for calculating edit distances in multicolored and directed graphs, extending existing theories with new methods involving random structures and graph homomorphisms.

Findings

Derived formulas for edit distances in multicolored graphs

Extended theory to directed graphs and hereditary properties

Provided bounds and algorithms for minimal edge recolorings

Abstract

The editing of a combinatorial object is the alteration of some of its elements such that the resulting object satisfies a certain fixed property. The edit problem for graphs, when the edges are added or deleted, was first studied independently by the authors and K\'ezdy [J. Graph Theory (2008), 58(2), 123--138] and by Alon and Stav [Random Structures Algorithms (2008), 33(1), 87--104]. In this paper, a generalization of graph editing is considered for multicolorings of the complete graph as well as for directed graphs. Specifically, the number of edge-recolorings sufficient to be performed on any edge-colored complete graph to satisfy a given hereditary property is investigated. The theory for computing the edit distance is extended using random structures and so-called types or colored homomorphisms of graphs.