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Characterizing Star-PCGs

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Abstract

A graph G is called a pairwise compatibility graph (PCG, for short) if it admits a tuple \((T,w, d_{\min },d_{\max })\) of a tree T whose leaf set is equal to the vertex set of G, a non-negative edge weight w, and two non-negative reals \(d_{\min }\le d_{\max }\) such that G has an edge between two vertices \(u,v\in V\) if and only if the distance between the two leaves u and v in the weighted tree (T, w) is in the interval \([d_{\min }, d_{\max }]\). The tree T is also called a witness tree of the PCG G. How to recognize PCGs is a wide-open problem in the literature. This paper gives a complete characterization for a graph to be a star-PCG (a PCG that admits a star as its witness tree), which provides us the first polynomial-time algorithm for recognizing star-PCGs.

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References

  1. Booth, S., Lueker, S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13, 335–379 (1976)

    Article  MathSciNet  Google Scholar 

  2. Brandstädt, A.: On leaf powers. Technical report, University of Rostock (2010)

  3. Calamoneri, T., Frascaria, D., Sinaimeri, B.: All graphs with at most seven vertices are pairwise compatibility graphs. Comput. J. 56(7), 882–886 (2013)

    Article  Google Scholar 

  4. Calamoneri, T., Frangioni, A., Sinaimeri, B.: Pairwise compatibility graphs of caterpillars. Comput. J. 57(11), 1616–1623 (2014)

    Article  Google Scholar 

  5. Calamoneri, T., Petreschi, R.: On pairwise compatibility graphs having Dilworth number two. Theor. Comput. Sci. 524, 34–40 (2014)

    Article  MathSciNet  Google Scholar 

  6. Calamoneri, T., Petreschi, R.: On pairwise compatibility graphs having Dilworth number \(k\). Theor. Comput. Sci. 547, 82–89 (2014)

    Article  MathSciNet  Google Scholar 

  7. Calamoneri, T., Petreschi, R., Sinaimeri, B.: On the pairwise compatibility property of some superclasses of threshold graphs. Discrete Math. Algorithms Appl. 5(2), 360002 (2013)

    Article  MathSciNet  Google Scholar 

  8. Calamoneri, T., Sinaimeri, B.: Pairwise compatibility graphs: a survey. SIAM Rev. 58(3), 445–460 (2016)

    Article  MathSciNet  Google Scholar 

  9. Chen, Z.-Z., Jiang, T., Lin, G.: Computing phylogenetic roots with bounded degrees and errors. SIAM J. Comput. 32, 864–879 (2003)

    Article  MathSciNet  Google Scholar 

  10. Durocher, S., Mondal, D., Rahman, MdS: On graphs that are not PCGs. Theor. Comput. Sci. 571, 78–87 (2015)

    Article  MathSciNet  Google Scholar 

  11. Felsenstein, J.: Cases in which parsimony or compatibility methods will be positively misleading. Syst. Zool. 27, 401–410 (1978)

    Article  Google Scholar 

  12. Hossain, MdI, Salma, S.A., Rahman, MdS, Mondal, D.: A necessary condition and a sufficient condition for pairwise compatibility graphs. J. Graph Algorithms Appl. 21(3), 341–352 (2017)

    Article  MathSciNet  Google Scholar 

  13. Kearney, P.E., Munro, J.I., Phillips, D.: Efficient generation of uniform samples from phylogenetic trees. In: Benson, G., Page, R. (eds.) Algorithms in Bioinformatics, LNCS 2812, pp. 177–189. Springer, Berlin (2003)

    Google Scholar 

  14. Lin, G., Kearney, P.E., Jiang, T.: Phylogenetic k-root and Steiner k-root. In: Goos, G., Hartmanis, J., van Leeuwen, J., Lee, D.T., Teng, S.-H. (eds.) Algorithms and Computation, LNCS 1969, pp. 539–551. Springer, Berlin (2000)

    Google Scholar 

  15. Mehnaz, S., Rahman, M.S.: Pairwise compatibility graphs revisited. In: Proceedings of the 2013 International Conference on Informatics, Electronics Vision (ICIEV), pp. 1–6 (2013)

  16. Nishimura, N., Ragde, P., Thilikos, D.M.: On graph powers for leaf-labeled trees. J. Algorithms 42, 69–108 (2002)

    Article  MathSciNet  Google Scholar 

  17. Salma, S.A., Rahman, MdS, Hossain, MdI: Triangle-free outerplanar 3-graphs are pairwise compatibility graphs. J. Graph Algorithms Appl. 17, 81–102 (2013)

    Article  MathSciNet  Google Scholar 

  18. Xiao, M., Nagamochi, H.: Some reduction operations to pairwise compatibility graphs. Inf. Process. Lett. 153, 105875 (2020)

    Article  MathSciNet  Google Scholar 

  19. Yanhaona, M.N., Bayzid, MdS, Rahman, MdS: Discovering pairwise compatibility graphs. Discrete Math. Algorithm Appl. 2(4), 607–624 (2010)

    Article  MathSciNet  Google Scholar 

  20. Yanhaona, M.N., Hossain, K.S.M.T., Rahman, MdS: Pairwise compatibility graphs. J. Appl. Math. Comput. 30, 479–503 (2009)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The work was supported by the National Natural Science Foundation of China, under Grants 61972070 and 61772115.

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Authors and Affiliations

  1. School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China

    Mingyu Xiao

  2. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, Japan

    Hiroshi Nagamochi

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  1. Mingyu Xiao
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  2. Hiroshi Nagamochi
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Correspondence to Mingyu Xiao.

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Xiao, M., Nagamochi, H. Characterizing Star-PCGs. Algorithmica 82, 3066–3090 (2020). https://doi.org/10.1007/s00453-020-00712-8

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  • DOI: https://doi.org/10.1007/s00453-020-00712-8

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