Abstract
A graph \(G=(V,E)\) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers \(d_{min}\) and \(d_{max}\), \(d_{min} \le d_{max}\), such that each node \(u \in V\) is uniquely associated to a leaf of T and there is an edge \((u,v) \in E\) if and only if \(d_{min} \le d_{T} (u, v) \le d_{max}\), where \(d_{T} (u, v)\) is the sum of the weights of the edges on the unique path \(P_{T}(u,v)\) from u to v in T. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. Despite numerous efforts, a complete characterization of the PCG class is not known yet. In this paper we propose a new proof technique that allows us to show that some interesting classes of graphs have empty intersection with PCG. We demonstrate our technique by showing many graph classes that do not lie in PCG. As a side effect, we show a not pairwise compatibility planar graph with 8 nodes (i.e. \(C^2_8\)), so improving the previously known result concerning the smallest planar graph known not to be PCG.
Partially supported by Sapienza University of Rome projects “Graph Algorithms for Phylogeny: a promising approach” and “Combinatorial structures and algorithms for problems in co-phylogeny”.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Baiocchi, P., Calamoneri, T., Monti, A., Petreschi, R.: Some classes of graphs that are not PCGs. arXiv:1707.07436 [cs.DM]
Brandstädt, A., Hundt, C.: Ptolemaic graphs and interval graphs are leaf powers. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 479–491. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78773-0_42
Calamoneri, T., Frangioni, A., Sinaimeri, B.: Pairwise compatibility graphs of caterpillars. Comput. J. 57(11), 1616–1623 (2014)
Calamoneri, T., Frascaria, D., Sinaimeri, B.: All graphs with at most seven vertices are pairwise compatibility graphs. Comput. J. 56(7), 882–886 (2013)
Calamoneri, T., Sinaimeri, B.: On pairwise compatibility graphs: a survey. SIAM Rev. 58(3), 445–460 (2016)
Calamoneri, T., Petreschi, R.: On pairwise compatibility graphs having Dilworth number two. Theoret. Comput. Sci. 524, 34–40 (2014)
Durocher, S., Mondal, D., Rahman, Md.S.: On graphs that are not PCGs. Theoret. Comput. Sci. 571, 78–87 (2015)
Felsenstein, J.: Cases in which parsimony or compatibility methods will be positively misleading. Syst. Zool. 27, 401–410 (1978)
Kearney, P., Munro, J.I., Phillips, D.: Efficient generation of uniform samples from phylogenetic trees. In: Benson, G., Page, R.D.M. (eds.) WABI 2003. LNCS, vol. 2812, pp. 177–189. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-39763-2_14
Mehnaz, S., Rahman, M.S.: Pairwise compatibility graphs revisited. In: Proceedings of the International Conference on Informatics, Electronics Vision (ICIEV) (2013)
Salma, S.A., Rahman, Md.S.: Triangle-free outerplanar 3-graphs are pairwise compatibility graphs. J. Graph Algorithms Appl. 17(2), 81–102 (2013)
Yanhaona, M.N., Bayzid, Md.S., Rahman, Md.S.: Discovering pairwise compatibility graphs. Discrete Math. Algorithms Appl. 2(4), 607–623 (2010)
Yanhaona, M.N., Hossain, K.S.M.T., Rahman, Md.S.: Pairwise compatibility graphs. J. Appl. Math. Comput. 30, 479–503 (2009)
Hossain, Md.I., Salma, S.A., Rahman, Md.S., Mondal, D.: A necessary condition and a sufficient condition for pairwise compatibility graphs. J. Graph Algorithms Appl. 21(3), 341–352 (2017)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Baiocchi, P., Calamoneri, T., Monti, A., Petreschi, R. (2018). Graphs that Are Not Pairwise Compatible: A New Proof Technique (Extended Abstract). In: Iliopoulos, C., Leong, H., Sung, WK. (eds) Combinatorial Algorithms. IWOCA 2018. Lecture Notes in Computer Science(), vol 10979. Springer, Cham. https://doi.org/10.1007/978-3-319-94667-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-94667-2_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94666-5
Online ISBN: 978-3-319-94667-2
eBook Packages: Computer ScienceComputer Science (R0)