Betweenness in graphs: A short survey on shortest and induced path betweennessFootnote

Peer review under responsibility of Kalasalingam University.

Footnote

This work was supported by NBHM DAE under file NBHM/2/48(9)/2014/NBHM (R.P.)/R&D-II/4364.

References

• Moritz Pasch Vorlesungen über Neuere Geometrie 1882 Teubner Leipzig
   Google Scholar
• Pambuccian Victor Axiomatizations of hyperbolic and absolute geometries Non-Euclidean Geom. 2006 119 153
   Google Scholar
• Pambuccian Victor The axiomatics of ordered geometry: I. Ordered incidence spaces Expo. Math. 29 1 Special Issue: Special Issue: International Conference on Discrete Mathematics 2011 24 66
   Web of Science ®Google Scholar
• Giuseppe Peano, I principii di geometria logicamente esposti, Fratelli Bocca, 1889.
   Google Scholar
• Hilbert David Grundlagen der Geometrie, Vol. 7 1913 BG Teubner
   Google Scholar
• Sholander Marlow Trees, lattices, order, and betweenness Proc. Amer. Math. Soc. 3 3 1952 369 381
   Web of Science ®Google Scholar
• Sholander Marlow Medians, lattices, and trees Proc. Amer. Math. Soc. 5 5 1954 808 812
   Web of Science ®Google Scholar
• Simovici Dan A. Betweenness, metrics and entropies in lattices 38th International Symposium on Multiple Valued Logic, 2008 ISMVL 2008 (2008) IEEE. 26–31.
   Google Scholar
• Duntsch Ivo Urquhart Alasdair Betweenness and comparability obtained from binary relations Lecture Notes in Comput. Sci. 4136 2006 148 161
   Google Scholar
• Prenowitz Walter Jantosciak James Geometries and join spaces J. Reine Angew. Math. 257 1972 100 128
   Web of Science ®Google Scholar
• Hedlíková Jarmila Ternary spaces, media, and Chebyshev sets Czechoslovak Math. J. 33 3 1983 373 389
   Web of Science ®Google Scholar
• Hedlíková Jarmila Katriňák Tibor Lattice betweenness relation and a generalization of König’s lemma Math. Slovaca 46 4 1996 343 354
   Google Scholar
• E. Sampathkumar, B-systems, in: Graph Theory and its Applications, Proceedings of the National Workshop, Manonmaniam Sundaranar University, Tirunelbeli, 1996, pp. 173–185.
   Google Scholar
• Bandelt Hans-Jürgen Chepoi Victor The algebra of metric betweenness I: Subdirect representation and retraction European J. Combin. 28 6 2007 1640 1661
   Web of Science ®Google Scholar
• Bandelt Hans-Jürgen Chepoi Victor The algebra of metric betweenness II: Geometry and equational characterization of weakly median graphs European J. Combin. 29 3 2008 676 700
   Web of Science ®Google Scholar
• Mulder Henry Martyn The interval function of a graph MC Tracts 132 1980 1 191
   Google Scholar
• Nieminen Juhani Join space graphs J. Geom. 33 1 Special Issue: Special Issue: International Conference on Discrete Mathematics 1988 99 103
   Google Scholar
• Nebeskỳ Ladislav A characterization of the interval function of a connected graph Czechoslovak Math. J. 44 1 Special Issue: Special Issue: International Conference on Discrete Mathematics 1994 173 178
   Web of Science ®Google Scholar
• Nebeskỳ Ladislav A characterization of the set of all shortest paths in a connected graph Math. Bohem. 119 1 Special Issue: Special Issue: International Conference on Discrete Mathematics 1994 15 20
   Google Scholar
• Nebesky Ladislav A characterization of the interval function of a (finite or infinite) connected graph Czechoslovak Math. J. 51 3 2001 635 642
   Web of Science ®Google Scholar
• Nebeskỳ Ladislav The interval function of a connected graph and a characterization of geodetic graphs Math. Bohem. 126 1 Special Issue: Special Issue: International Conference on Discrete Mathematics 2001 247 254
   Google Scholar
• Nebeskỳ Ladislav Intervals and steps in a connected graph Discrete Math. 286 1 Special Issue: Special Issue: International Conference on Discrete Mathematics 2004 151 156
   Web of Science ®Google Scholar
• Nebeskỳ Ladislav The interval function of a connected graph and road systems Discrete Math. 307 16 2007 2067 2073
   Web of Science ®Google Scholar
• Mulder Henry Martyn Nebeskỳ Ladislav Axiomatic characterization of the interval function of a graph European J. Combin. 30 5 2009 1172 1185
   Web of Science ®Google Scholar
• Mulder Henry Martyn Transit Functions on Graphs (and Posets) Lecture Notes Series (2008) 117.
   Google Scholar
• Nebeskỳ Ladislav A characterization of the interval function of a connected graph Czechoslovak Math. J. 44 1 Special Issue: Special Issue: International Conference on Discrete Mathematics 1994 173 178
   Web of Science ®Google Scholar
• Mulder Henry Martyn Nebeskỳ Ladislav Axiomatic characterization of the interval function of a graph European J. Combin. 30 5 2009 1172 1185
   Web of Science ®Google Scholar
• Changat Manoj Nezhad Ferdoos Hossein Mulder Henry Martyn Narayanan N. A note on the interval function of a disconnected graph Discuss. Math. Graph Theory 2017
   Web of Science ®Google Scholar
• Chepoi Victor Separation of two convex sets in convexity structures J. Geom. 50 1 Special Issue: Special Issue: International Conference on Discrete Mathematics 1994 30 51
   Google Scholar
• Victor Chepoi, Kolja Knauer, Tilen Marc, Partial cubes without Q3− minors, 2016. arXiv preprint http://arxiv:1606.02154.
   Google Scholar
• Djoković D.Ž. Distance-preserving subgraphs of hypercubes J. Combin. Theory Ser. B 14 3 1973 263 267
   Google Scholar
• Albenque Marie Knauer Kolja Convexity in partial cubes: the hull number Discrete Math. 339 2 2016 866 876
   Web of Science ®Google Scholar
• Bandelt Hans-Jürgen Graphs with intrinsic S3 convexities J. Graph Theory 13 2 1989 215 228
   Web of Science ®Google Scholar
• Chepoi V.D. d-Convex Sets in Graphs (Ph.D. thesis, Ph.D. dissertation) 1986 Moldova State University Kishinev (Russian)
   Google Scholar
• Chepoi Victor Classifying graphs by metric triangles Metody Diskret. Anal. 49 1989 75 93 (Russian)
   Google Scholar
• Changat Manoj Mathew Joseph Mulder Henry Martyn The induced path function, monotonicity and betweenness Discrete Appl. Math. 158 5 2010 426 433
   Web of Science ®Google Scholar
• Changat Manoj Lakshmikuttyamma Anandavally K. Mathews Joseph Peterin Iztok Narasimha-Shenoi Prasanth G. Seethakuttyamma Geetha Špacapan Simon A forbidden subgraph characterization of some graph classes using betweenness axioms Discrete Math. 313 8 2013 951 958
   Web of Science ®Google Scholar
• Sholander Marlow Trees, lattices, order, and betweenness Proc. Amer. Math. Soc. 3 3 1952 369 381
   Web of Science ®Google Scholar
• Chvátal Vašek Rautenbach Dieter Schäfer Philipp Matthias Finite Sholander trees, trees, and their betweenness Discrete Math. 311 20 2011 2143 2147
   Web of Science ®Google Scholar
• Balakrishnan Kannan Changat Manoj Lakshmikuttyamma Anandavally K. Mathew Joseph Mulder Henry Martyn Narasimha-Shenoi Prasanth G. Narayanan N. Axiomatic characterization of the interval function of a block graph Discrete Math. 338 6 2015 885 894
   Web of Science ®Google Scholar
• Changat Manoj Nezhad Ferdoos Hossein Narayanan Narayanan Axiomatic characterization of the interval function of a bipartite graph Conference on Algorithms and Discrete Applied Mathematics 2017 Springer 96 106
   Google Scholar
• Changat Manoj Nezhad Ferdoos Hossein Narayanan N. Axiomatic characterization of claw and paw-free graphs using graph transit functions Conference on Algorithms and Discrete Applied Mathematics 2016 Springer 115 125
   Google Scholar
• Nebeskỳ Ladislav The induced paths in a connected graph and a ternary relation determined by them Math. Bohem. 127 3 2002 397 408
   Google Scholar
• Morgana Maria Aurora Mulder Henry Martyn The induced path convexity, betweenness, and svelte graphs Discrete Math. 254 1–3 2002 349 370
   Web of Science ®Google Scholar
• Changat Manoj Mathew J. Characterizations of J-monotone graphs Convexity Discrete Struct. 5 2008 47 55
   Google Scholar
• Changat Manoj Peterin Iztok Ramachandran Abisha Tepeh Aleksandra The induced path transit function and the Pasch axiom Bull. Malaysian Math. Sci. Soc. (2) 39 1 Special Issue: Special Issue: International Conference on Discrete Mathematics 2016 123 134
   Google Scholar
• Changat Manoj Klavzar Sandi Mulder Henry Martyn The all-paths transit function of a graph Czechoslovak Math. J. 51 2 2001 439 448
   Web of Science ®Google Scholar