The Longest Path Transit Function of a graph and Betweenness

A longest path between two vertices in a connected graph G is a path of maximum length between the vertices. The longest path transit function L (u, v) in a graph consists of the set of all vertices lying on any longest path between vertices u and v. A transit function L satisfies betweenness if w is an element of L(u, v) implies u is not an element of L(w, v) [(b1)-axiom] and x is an element of L(u, v) implies L(u, x) subset of L(u, v) [(b2)-axiom] and it is monotone if x, y is an element of L(u, v) implies L(x, y) subset of L(u, v). The betweenness and monotone axioms are discussed for the longest path transit function of G. Some graphs are identified for L to become a single path transit function..