Longest path transit function of a graph References - Fmf

Longest path transit function of a graph Manoj Changat, 2007 Definition(s): A transit function on a set V is a function R:V ×V → 2V satisfying the following axioms for any u, v in V : (t1) u ∈ R(u, v), (t2) R(u, v) = R(v, u), (t3) R(u, u) = {u}. If G = (V, E) is a connected graph and if R is a transit function on V then we say that R is a transit function on G. Given a transit function R on G, define R(u, v, w) = R(u, v) ∩ R(v, w) ∩ R(u, w). An interesting problem is to study R(u, v, w), for each triple of vertices (u, v, w) in G. For example which graphs is R(u, v, w) 6= ∅ or which graphs is |R(u, v, w)| = 1, for any triple of vertices (u, v, w). (See [1] and [3], for details about transit functions on graphs.) 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. It can be verified easily that the length of a longest path between two vertices is also a metric on V . Problem: For any triple of vertices (u, v, w) in a connected graph G, is L(u, v, w) 6= ∅? In addition, characterize graphs for which |L(u, v, w)| = 1. Variations: The graphs for which this holds when R is the geodesic transit function I are precisely the modular graphs and median graphs, respectively. Analogous question for the induced path transit function J is solved by Morgano and Mulder [2].

References [1] M. Changat, H.M. Mulder, G. Sierksma, Convexities related to path properties on graphs, Discrete Math. 290 (2005) 117–131. [2] M.A. Morgana, H.M. Mulder, The induced path convexity, betweenness and svelte graphs. Discrete Math. 254 (2002) 349–370. [3] H.M. Mulder, Transit functions on graphs, to appear.