Convexities related to path properties on graphs

A feasible family of paths in a connected graph G is a family that contains at least one path between any pair of vertices in G. Any feasible path family defines a convexity on G. Well-known instances are: the geodesics, the induced paths, and all paths. We propose a more general approach for such 'path properties'. We survey a number of results from this perspective, and present a number of new results. We focus on the behaviour of such convexities on the Cartesian product of graphs and on the classical convexity invariants, such as the Caratheodory, Helly and Radon numbers in relation with graph invariants, such as the clique number and other graph properties.

[1]  Robert E. Jamison-Waldner.  PARTITION NUMBERS FOR TREES AND ORDERED SETS , 1981 .

[2]  H. M. Mulder.  The interval function of a graph , 1980 .

[3]  G. Sierksma.  CARATHEODORY AND HELLY-NUMBERS OF CONVEX-PRODUCT-STRUCTURES , 1975 .

[4]  Gary Chartrand,et al.  Convex sets in graphs , 1999 .

[5]  Henry Martyn Mulder,et al.  The induced path convexity, betweenness, and svelte graphs , 2002, Discret. Math..

[6]  Ladislav Nebeský.  A characterization of the set of all shortest paths in a connected graph , 1994 .

[7]  J. Eckhoff.  Helly, Radon, and Carathéodory Type Theorems , 1993 .

[8]  M. Farber,et al.  Convexity in graphs and hypergraphs , 1986 .

[9]  Ladislav Nebeský.  Characterizing the interval function of a connected graph , 1998 .

[10]  Ladislav Nebeský,et al.  A characterization of the interval function of a connected graph , 1994 .

[11]  P. Duchet,et al.  Discrete convexity : retractions , morphisms and the partition problem , 1998 .

[12]  Van de M. L. J. Vel.  Theory of convex structures , 1993 .

[13]  Pierre Duchet.  Convexity in combinatorial structures , 1987 .

[14]  J. Calder.  Some Elementary Properties of Interval Convexities , 1971 .

[15]  Pierre Duchet,et al.  Convex sets in graphs, II. Minimal path convexity , 1987, J. Comb. Theory B.

[16]  Jörg M. Wills,et al.  Handbook of Convex Geometry , 1993 .

[17]  Manoj Changat,et al.  On triangle path convexity in graphs , 1999, Discret. Math..

[18]  Sandi Klavzar,et al.  The All-Paths Transit Function of a Graph , 2001 .