Abstract
Let T be an edge weighted tree and let \(d_{min}\), \(d_{max}\) be two non-negative real numbers where \(d_{min}\le d_{max}\). A pairwise compatibility graph (PCG) of T for \(d_{min}\), \(d_{max}\) is a graph G such that each vertex of G corresponds to a distinct leaf of T and two vertices are adjacent in G if and only if the weighted distance between their corresponding leaves lies within the interval \([d_{min},d_{max}]\). A graph G is a PCG if there exist an edge weighted tree T and suitable \(d_{min}\), \(d_{max}\) such that G is a PCG of T. Knowing that all graphs are not PCGs, in this paper we introduce a variant of pairwise compatibility graphs which we call multi-interval PCGs. A graph G is a multi-interval PCG if there exist an edge weighted tree T and some mutually exclusive intervals of nonnegative real numbers such that there is an edge between two vertices in G if and only if the distance between their corresponding leaves in T lies within any such intervals. If the number of intervals is k, then we call the graph a k-interval PCG. We show that every graph is a k-interval PCG for some k. We also prove that wheel graphs and a restricted subclass of series-parallel graphs are 2-interval PCGs.
Keywords
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Battista, G.D., Tamassia, R.: On-line maintenance of triconnected components with SPQR-trees. Algorithmica 15(4), 302–318 (1996)
Battista, G.D., Tamassia, R., Vismara, L.: Output-sensitive reporting of disjoint paths. Algorithmica 23(4), 302–340 (1999)
Bayzid, M.S.: On Pairwise compatibility graphs. Master’s thesis, Bangladesh University of Engineering and Technology, June 2010
Calamoneri, T., Frascaria, D., Sinaimeri, B.: All graphs with at most seven vertices are pairwise compatibility graphs. Comput. J. 57, 882–886 (2013)
Calamoneri, T., Petreschi, R., Sinaimeri, B.: On relaxing the constraints in pairwise compatibility graphs. In: Rahman, M.S., Nakano, S. (eds.) WALCOM 2012. LNCS, vol. 7157, pp. 124–135. Springer, Heidelberg (2012). doi:10.1007/978-3-642-28076-4_14
Calamoneri, T., Petreschi, R., Sinaimeri, B.: On the pairwise compatibility property of some superclasses of threshold graphs. Discrete Math. Algorithms Appl. 5(2) (2013)
Durocher, S., Mondal, D., Rahman, M.S.: On graphs that are not PCGs. Theor. Comput. Sci. 571, 78–87 (2015)
Hossain, M.I., Salma, S.A., Rahman, M.S.: A necessary condition and a sufficient condition for pairwise compatibility graphs. In: Kaykobad, M., Petreschi, R. (eds.) WALCOM 2016. LNCS, vol. 9627, pp. 107–113. Springer, Cham (2016). doi:10.1007/978-3-319-30139-6_9
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). doi:10.1007/978-3-540-39763-2_14
Rahman, M.S., Egi, N., Nishizeki, T.: No-bend orthogonal drawings of series-parallel graphs. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 409–420. Springer, Heidelberg (2006). doi:10.1007/11618058_37
Salma, S.A., Rahman, M.S., Hossain, M.I.: Triangle-free outerplanar 3-graphs are pairwise compatibility graphs. J. Graph Algorithms Appl. 17(2), 81–102 (2013)
Yanhaona, M.N., Bayzid, M.S., Rahman, M.S.: Discovering pairwise compatibility graphs. Discrete Math. Algorithms Appl. 2, 607–623 (2010)
Yanhaona, M.N., Hossain, K.S.M.T., Rahman, M.S.: Pairwise compatibility graphs. J. Appl. Math. Comput. 30, 479–503 (2009)
Acknowledgments
We thank Kazuo Iwama of Kyoto University who pointed out this variant of the problem when the second author discussed the PCG problem with him in 2014.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Ahmed, S., Rahman, M.S. (2017). Multi-interval Pairwise Compatibility Graphs. In: Gopal, T., Jäger , G., Steila, S. (eds) Theory and Applications of Models of Computation. TAMC 2017. Lecture Notes in Computer Science(), vol 10185. Springer, Cham. https://doi.org/10.1007/978-3-319-55911-7_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-55911-7_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-55910-0
Online ISBN: 978-3-319-55911-7
eBook Packages: Computer ScienceComputer Science (R0)