Prof. Dieter Rautenbach
Universität Ulm

The Convexity Space induced by Paths of Order Three

Every set P of paths of a graph G defines a natural convexity space where some set C of vertices of G is considered convex exactly if for every path p in P that starts and ends in C, the set C contains all vertices of p. The well-known geodetic convexity and monophonic convexity for example are obtained in this way considering either the set of all shortest paths or the set of all induced paths of G. In this talk we consider as P the set of all paths of G of order 3, which leads to a convexity notion that was considered in connection with rumour/desease spreading processes in graphs. We survey several of our recent results concerning the classical parameters associated with convexity spaces, namely the hull number, the Caratheodory number and the Radon number. The presented material based on three papers and joint work with several coauthors.