Chen and Chvatal's conjecture in tournaments

In a directed graph D, given two distinct vertices u and v, the line defined by the ordered pair (u, v) is the set of all vertices w such that u, v and w belong to a shortest directed path in D, containing a shortest directed path from u to v. In this work we study the following conjecture: the number of distinct lines in any strongly connected graph is at least its number of vertices, unless there is a line containing all the vertices. Our main result is that any tournament satisfies this conjecture; we also prove this for bipartite tournaments of diameter at most three. (C) 2021 Elsevier Ltd. All rights reserved.