Large-girth roots of graphs

Details Large-girth roots of graphs Large-girth roots of graphs

2009-09-22

arxiv.org/abs/0909.4011v1

Computer Science

Data Structures and Algorithms

14 pages, 4 figures

Scientific paper

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.

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