  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.
