Hardness Results and Efficient Algorithms for Graph Powers
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 ≥ 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 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 − 1 is polynomially recognizable.
KeywordsBipartite Graph Maximal Clique Chordal Graph Hardness Result Split Graph
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- 4.Farzad, B., Lau, L.C., Le, V.B., Nguyen, N.T.: Computing graph roots without short cycles. In: Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS 2009), pp. 397–408 (2009)Google Scholar
- 12.Lubiw, A.: Γ-free matrices, Master Thesis, Dept. of Combinatorics and Optimization, University of Waterloo, Canada (1982)Google Scholar