Computing roots of graphs is hard
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Computing square roots of trivially perfect and threshold graphs
Discrete Applied Mathematics
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We study the problem of recognizing graph powers and computing roots of graphs. Our focus is on classes of graphs with no short cycles. We provide a polynomial time recognition algorithm for $r$-th powers of graphs of girth at least $2r+3$, thus improving a recently conjectured bound. 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 toward a recent conjecture of Levenshtein [Discrete Math., 308 (2008), pp. 993-998] that such roots should be unique. Similar algorithms have so far been designed only for $r=2,3$. On the negative side, we prove that recognition of graph powers becomes an NP-complete problem when the bound on girth is about twice smaller. (Anna Adamaszek's correct affiliation is Department of Computer Science and Centre for Discrete Mathematics and Its Applications (DIMAP), University of Warwick, Coventry, CV4 7AL, UK.)