The longest path transit function of a graph and betweenness

Abstract

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 ∈ L(u, v) implies u ∉ L(w,v) [(b1)-axiom] and x ∈ L(u,v) implies L(u, x) ⊆ L(u, v) [(b2)-axiom] and it is monotone if x, y e L(u, v) implies L(x, y) ⊆ 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.