Levenshtein Graphs: Resolvability, Automorphisms & Determining Sets

TL;DR

This paper introduces Levenshtein graphs based on edit distance, characterizes their properties, automorphisms, and metric dimension, and provides algorithms for computing edit distances efficiently.

Contribution

It defines Levenshtein graphs with variable string lengths, characterizes their automorphisms and resolving sets, and presents an efficient edit distance algorithm.

Findings

Characterized when geodesic distance equals edit distance.

Determined automorphism groups and resolving sets.

Developed an O(k) algorithm for edit distance to specific strings.

Abstract

We introduce the notion of Levenshtein graphs, an analog to Hamming graphs but using the edit distance instead of the Hamming distance; in particular, Levenshtein graphs allow for underlying strings (nodes) of different lengths. We characterize various properties of these graphs, including a necessary and sufficient condition for their geodesic distance to be identical to the edit distance, their automorphism group and determining number, and an upper bound on their metric dimension. Regarding the latter, we construct a resolving set composed of two-run strings and an algorithm that computes the edit distance between a string of length $k$ and any single-run or two-run string in $O(k)$ operations.