Hardness Results and Efficient Algorithms for Graph Powers

The k-th power H 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 ≥ 2 and recognizing k-th powers of a bipartite graph is NP-complete for all fixed k ≥ 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 ≥ 2 and providing a quadratic-time recognition algorithm for squares of strongly chordal split graphs. Finally, we give a polynomialtime 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−1 is polynomially recognizable.