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Ptolemaic and Planar Cover-incomparability Graphs

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Abstract

Cover-Incomparability graphs (C-I graphs) are graphs whose edge-set is the union of edge-sets of incomparability graph and the cover graph of some poset. We give a new characterization of Ptolemaic C-I graphs and prove that each C-I graph contains a Ptolemaic C-I graph as a spanning subgraph. This result is used to present several necessary conditions for a graph G being planar C-I graph. We present a hierarchy of subfamilies of chordal graphs in the class of C-I graphs and prove that many different graph families coincide in the class of C-I graphs.

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Data Availability Statement

Data sharing not applicable to this article as no data sets were generated or analysed during the current study, certified that this article is theoretical and contains no data sets that were generated or analysed during the current study.

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Acknowledgments

A.A. acknowledges the financial support from UGC Govt. of India for providing Junior Research Fellowship. M.C. acknowledges the financial support from SERB, Department of Science & Technology, Govt. of India (research project under MATRICS scheme No. MTR/2017/000238). The financial support from the Slovenian Research Agency (research core funding P1-0297, and projects J1-9109, J1-1693) is acknowledged by T.G.

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Authors and Affiliations

  1. Department of Futures Studies, University of Kerala, Thiruvananthapuram, 695581, India

    Arun Anil & Manoj Changat

  2. Faculty of natural sciences and mathematics, University of Maribor and Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Tanja Gologranc

  3. Department of Mathematics, S.N. College, Punalur, 691305, India

    Baiju Sukumaran

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  1. Arun Anil
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  2. Manoj Changat
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  3. Tanja Gologranc
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  4. Baiju Sukumaran
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Correspondence to Manoj Changat.

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Anil, A., Changat, M., Gologranc, T. et al. Ptolemaic and Planar Cover-incomparability Graphs. Order 38, 421–439 (2021). https://doi.org/10.1007/s11083-021-09549-4

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