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Complexity of Finding Graph Roots with Girth Conditions

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Abstract

Graph G is the square of graph H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Given H it is easy to compute its square H 2, however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H (of girth 3). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if G=H 2 for some graph H of small girth. The main results are the following.

  • There is a graph theoretical characterization for graphs that are squares of some graph of girth at least 7. A corollary is that if a graph G has a square root H of girth at least 7 then H is unique up to isomorphism.

  • There is a polynomial time algorithm to recognize if G=H 2 for some graph H of girth at least 6.

  • It is NP-complete to recognize if G=H 2 for some graph H of girth 4.

These results almost provide a dichotomy theorem for the complexity of the recognition problem in terms of girth of the square roots. The algorithmic and graph theoretical results generalize previous results on tree square roots, and provide polynomial time algorithms to compute a graph square root of small girth if it exists. Some open questions and conjectures will also be discussed.

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Authors and Affiliations

  1. Department of Mathematics, Brock University, Saint Catharines, Canada

    Babak Farzad

  2. Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong, Hong Kong

    Lap Chi Lau

  3. Institut für Informatik, Universität Rostock, Rostock, Germany

    Van Bang Le

  4. Department of Computer Science, Hong Duc University, Thanh Hoa City, Vietnam

    Nguyen Ngoc Tuy

Authors
  1. Babak Farzad
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  2. Lap Chi Lau
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  3. Van Bang Le
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  4. Nguyen Ngoc Tuy
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Corresponding author

Correspondence to Babak Farzad.

Additional information

Nguyen Ngoc Tuy is supported by the Ministry of Education and Training, Vietnam, Grant No. 3766/QD-BGD & DT.

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Farzad, B., Lau, L.C., Le, V.B. et al. Complexity of Finding Graph Roots with Girth Conditions. Algorithmica 62, 38–53 (2012). https://doi.org/10.1007/s00453-010-9442-9

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  • DOI: https://doi.org/10.1007/s00453-010-9442-9

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