Large-girth roots of graphs

TL;DR

This paper presents a polynomial time algorithm for recognizing and finding roots of graph powers with large girth, improving previous bounds, and explores the computational complexity when girth bounds are smaller.

Contribution

It introduces a polynomial time recognition algorithm for r-th powers of graphs with girth at least 2r+3 and analyzes the complexity when girth bounds are reduced.

Findings

Polynomial time recognition for graphs with girth ≥ 2r+3

Algorithm finds all such roots without degree one vertices

Recognition becomes NP-complete when girth bounds are smaller

Abstract

We study the problem of recognizing graph powers and computing roots of graphs. We provide a polynomial time recognition algorithm for r-th powers of graphs of girth at least 2r+3, thus improving a bound conjectured by Farzad et al. (STACS 2009). Our algorithm also finds all r-th roots of a given graph that have girth at least 2r+3 and no degree one vertices, which is a step towards a recent conjecture of Levenshtein that such root should be unique. On the negative side, we prove that recognition becomes an NP-complete problem when the bound on girth is about twice smaller. Similar results have so far only been attempted for r=2,3.