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 $r=2,3$.
@InProceedings{adamaszek_et_al:LIPIcs.STACS.2010.2442, author = {Adamaszek, Anna and Adamaszek, Michal}, title = {{Large-Girth Roots of Graphs}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {35--46}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2442}, URN = {urn:nbn:de:0030-drops-24429}, doi = {10.4230/LIPIcs.STACS.2010.2442}, annote = {Keywords: Graph roots, Graph powers, NP-completeness, Recognition algorithms} }
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