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Hardness Results and Efficient Algorithms for Graph Powers

  • Van Bang Le
  • Ngoc Tuy Nguyen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5911)

Abstract

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.

Keywords

Bipartite Graph Maximal Clique Chordal Graph Hardness Result Split Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Van Bang Le
    • 1
  • Ngoc Tuy Nguyen
    • 1
    • 2
  1. 1.Institut für InformatikUniversität RostockRostockGermany
  2. 2.Hong Duc UniversityThanh HoaVietnam

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