SIAM Journal on Discrete Mathematics
Computing square roots of trivially perfect and threshold graphs
Discrete Applied Mathematics
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The k-th power H k of a graph H is obtained from H by adding new edges between every two distinct vertices having distance at most k in H. Lau [Bipartite roots of graphs, ACM Transactions on Algorithms 2 (2006) 178---208] conjectured that recognizing k-th powers of some graph is NP-complete for all fixed k 驴 2 and recognizing k-th powers of a bipartite graph is NP-complete for all fixed k 驴 3. We prove that these conjectures are true. Lau and Corneil [Recognizing powers of proper interval, split and chordal graphs, SIAM J. Discrete Math. 18 (2004) 83---102] proved that recognizing squares of chordal graphs and squares of split graphs are NP-complete. We extend these results by showing that recognizing k-th powers of chordal graphs is NP-complete for all fixed k 驴 2 and providing a quadratic-time recognition algorithm for squares of strongly chordal split graphs. Finally, we give a polynomial-time recognition algorithm for cubes of graphs with girth at least ten. This result is related to a recent conjecture posed by Farzad et al. [Computing graph roots without short cycles, Proceedings of STACS 2009, pp. 397---408] saying that k-th powers of graphs with girth at least 3k 驴 1 is polynomially recognizable.