Transit functions on graphs (and posets)

The notion of transit function is introduced to present a unifying approach for results and ideas on intervals, convexities and betweenness in graphs and posets. Prime examples of such transit functions are the interval function I and the induced path function J of a connected graph. Another transit function is the all-paths function. New transit functions are introduced, such as the cutvertex transit function and the longest path function. The main idea of transit functions is that of ‘transferring’ problems and ideas of one transit function to the other. For instance, a result on the interval function I might suggest similar problems for the induced path function J. Examples are given of how fruitful this transfer can be. A list of Prototype Problems and Questions for this transferring process is given, which suggests many new questions and open problems.