Large-Girth Roots of Graphs

Anna Adamaszek, MichaŁ Adamaszek
2010 SIAM Journal on Discrete Mathematics  
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
more » ... ition 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. 1998 ACM Subject Classification: G.2.2 Graph algorithms, F.2.2 Analysis of algorithms and problem complexity.
doi:10.1137/100792949 fatcat:aqgpi37h5faihn6wvxodl624ay