Abstract
The Pasch axiom is a strong geometric property which was noted and discussed even from the period of Euclid. Modern geometers followed the study of the Pasch axiom, and in the context of axiomatic convexity, this axiom was widely studied by pioneers in convexity theory. Connected graphs for which the geodesic interval function I satisfies the Pasch axiom were characterized by Chepoi in 1994. In this paper, we characterize all graphs for which the induced path transit function satisfies the Pasch axiom. Among arbitrary transit functions R that satisfy the Pasch axiom, we identify the induced path transit function of the underlying graph \(G_R\) by a set of axioms.
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References
Bandelt, H.-J., Mulder, H.M.: Distance-hereditary graphs. J. Comb. Theory Ser. B 41, 182–208 (1986)
Changat, M., Mathew, J.: Interval monotone graphs: minimal path convexity. In: Balakrishnan, R., Mulder, H.M., Vijayakumar, A. (eds.) Proceedings of the Conference on Graph Connections, pp. 87–90, New Delhi (1999)
Changat, M., Mathew, J.: Induced path transit function, monotone and Peano axioms. Discrete Math. 286, 185–194 (2004)
Changat, M., Mathews, J., Mulder, H.M.: The induced path function, monotonicity and betweenness. Discrete Appl. Math. 158, 426–433 (2010)
Changat, M., Lakshmikuttyammaa, A.K., Mathews, J., Peterin, I., Narasimha-Shenoi, P.G., Seethakuttyammae, G., Špacapan, S.: A forbidden subgraph characterization of some graph classes using betweenness axioms. Discrete Math. 313, 951–958 (2013)
Chepoi, V.: Some properties of domain finite convexity structures (in Russian). Res. Algebra, Geometry and Appl. (Moldova State University), 142–148 (1986)
Chepoi, V.: Separation of two convex sets in convexity structures. J. Geom. 50, 30–51 (1994)
Duchet, P.: Convex sets in graphs II. Minimal path convexity. J. Comb. Theory Ser. B 44, 307–316 (1988)
Ellis, J.W.: A general set-separation theorem. Duke Math. J. 19, 417–421 (1952)
Morgana, M.A., Mulder, H.M.: The induced path convexity, betweenness and svelte graphs. Discrete Math. 254, 349–370 (2002)
Mulder, H.M.: Transit functions on graphs (and posets). In: Changat, M., Klavžar, S., Mulder, H.M., Vijayakumar, A. (eds.) Convexity in Discrete Structures, pp. 117–130, Lecture Notes Ser. 5, Ramanujan Math. Soc. (2008)
Nebeský, L.: A characterization of the interval function of a connected graph. Czechoslov. Math. J. 44, 173–178 (1994)
Nebeský, L.: The induced paths in a connected graph and a ternary relation determined by them. Math. Bohem. 127, 397–408 (2002)
Nieminen, J.: Join space graphs. J. Geom. 33, 99–103 (1988)
van de Vel, M.L.J.: Theory of Convex Structures. North Holland, Amsterdam (1993)
Acknowledgments
Work supported by the Ministry of Science of Slovenia and by the Ministry of Science and Technology of India under the bilateral India-Slovenia grants BI-IN/06-07-002 and DST/INT/SLOVENIA/P-17/2009, respectively. The first author was partially supported by NBHM-DAE, Govt. of India, under the research Grant No. 2/48(9)/2014/NBHM(R.P) R&D II/4364. The third author was partially supported by UGC, Govt. of India, under the research Grant No. F-17-57/09 (SA-I). The last author was partially supported by Slovenian Research Agency ARRS, Program No. P1–00383, Project No. L1–4292, and Creative Core–FISNM–3330-13-500033.
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Communicated by Sanming Zhou.
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Changat, M., Peterin, I., Ramachandran, A. et al. The Induced Path Transit Function and the Pasch Axiom. Bull. Malays. Math. Sci. Soc. 39 (Suppl 1), 123–134 (2016). https://doi.org/10.1007/s40840-015-0285-z
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DOI: https://doi.org/10.1007/s40840-015-0285-z