PDF
Discussiones Mathematicae Graph Theory 30(4) (2010)
671-685
DOI: https://doi.org/10.7151/dmgt.1522
n-ary TRANSIT FUNCTIONS IN GRAPHS
|
Manoj Changat, Joseph Mathews
Department of Futures Studies
| Iztok Peterin
Institute of Mathematics and Physics, FEECS | Prasanth G. Narasimha-Shenoi
Department of Mathematics, Government College, Chittur |
Abstract
n-ary transit functions are introduced as a generalization of binary (2-ary) transit functions. We show that they can be associated with convexities in natural way and discuss the Steiner convexity as a natural n-ary generalization of geodesicaly convexity. Furthermore, we generalize the betweenness axioms to n-ary transit functions and discuss the connectivity conditions for underlying hypergraph. Also n-ary all paths transit function is considered.Keywords: n-arity, transit function, betweenness, Steiner convexity.
2000 Mathematics Subject Classification: 52A01, O5C12.
References
| [1] | B. Bresar, M. Changat, J. Mathews, I. Peterin, P.G. Narasimha-Shenoi and A. Tepeh Horvat, Steiner intervals, geodesic intervals, and betweenness, Discrete Math. 309 (2009) 6114-6125, doi: 10.1016/j.disc.2009.05.022. |
| [2] | M. Changat, S. Klavžar and H.M. Mulder, The All-Paths Transit Function of a Graph, Czechoslovak Math. J. 51 (126) (2001) 439-448. |
| [3] | M. Changat and J. Mathew, Induced path transit function, monotone and Peano axioms, Discrete Math. 286 (2004) 185-194, doi: 10.1016/j.disc.2004.02.017. |
| [4] | M. Changat and J. Mathew, Characterizations of J-monotone graphs, in: Convexity in Discrete Structures (M. Changat, S. Klavžar, H.M. Mulder, A. Vijayakumar, eds.), Lecture Notes Ser. 5, Ramanujan Math. Soc. (2008) 47-55. |
| [5] | M. Changat, J. Mathew and H.M. Mulder, Induced path function, betweenness and monotonicity, Discrete Appl. Math. 158 (2010) 426-433, doi: 10.1016/j.dam.2009.10.004. |
| [6] | M. Changat, J. Mathew and H.M. Mulder, Induced path transit function, betweenness and monotonicity, Elect. Notes Discrete Math. 15 (2003). |
| [7] | M. Changat, H.M. Mulder and G. Sierksma, Convexities Related to Path Properties on Graphs, Discrete Math. 290 (2005) 117-131, doi: 10.1016/j.disc.2003.07.014. |
| [8] | M. Changat, P.G. Narasimha-Shenoi and I.M. Pelayo, The longest path transit function and its betweenness, to appear in Util. Math. |
| [9] | P. Duchet, Convexity in combinatorial structures, Rend. Circ. Mat. Palermo (2) Suppl. 14 (1987) 261-293. |
| [10] | P. Duchet, Convex sets in graphs II. Minimal path convexity, J. Combin. Theory (B) 44 (1988) 307-316, doi: 10.1016/0095-8956(88)90039-1. |
| [11] | P. Duchet, Discrete convexity: retractions, morphisms and partition problem, in: Proceedings of the conference on graph connections, India, (1998), Allied Publishers, New Delhi, 10-18. |
| [12] | P. Hall, On representation of subsets, J. Lon. Mat. Sc. 10 (1935) 26-30, doi: 10.1112/jlms/s1-10.37.26. |
| [13] | M.A. Morgana and H.M. Mulder, The induced path convexity, betweenness and svelte graphs, Discrete Math. 254 (2002) 349-370, doi: 10.1016/S0012-365X(01)00296-5. |
| [14] | H.M. Mulder, The Interval Function of a Graph. Mathematical Centre Tracts 132, Mathematisch Centrum (Amsterdam, 1980). |
| [15] | H.M. Mulder, Transit functions on graphs (and posets), in: Convexity in Discrete Structures (M. Changat, S. Klavžar, H.M. Mulder, A. Vijayakumar, eds.), Lecture Notes Ser. 5, Ramanujan Math. Soc. (2008) 117-130. |
| [16] | L. Nebeský, A characterization of the interval function of a connected graph, Czechoslovak Math. J. 44(119) (1994) 173-178. |
| [17] | L. Nebeský, A Characterization of the interval function of a (finite or infinite) connected graph, Czechoslovak Math. J. 51(126) (2001) 635-642. |
| [18] | E. Sampathkumar, Convex sets in graphs, Indian J. Pure Appl. Math. 15 (1984) 1065-1071. |
| [19] | M.L.J. van de Vel, Theory of Convex Structures (North Holland, Amsterdam, 1993). |
Received 11 November 2009
Accepted 2 March 2010
Close