Cycle transit function and betweenness

Transit functions are introduced to study betweenness, intervals and convexity in an axiomatic setup on graphs and other discrete structures. Prime example of a transit function on graphs is the well studied interval function of a connected graph. In this paper, we study the Cycle transit function $mathcal{C}( u,v)$ on graphs which is a transit function derived from the interval function. We study the betweenness properties and also characterize graphs in which the cycle transit function coincides with the interval function. We also characterize graphs where $|mathcal{C}( u,v)cap mathcal{C}( v,w) cap mathcal{C}( u,w)|le 1$ as an analogue of median graphs.