Mathematics — Graph Theory · tournaments-dominance
Problem
Let D be a tournament with at least one vertex.
We say that a vertex u of D directly owns a vertex w of D if (u,w) is an arc of D.
We say that a vertex u of D indirectly owns a vertex w of D if there exists a vertex v of D such that both (u,v) and (v,w) are arcs of D.
Prove that D has a vertex that (directly or indirectly) owns all other vertices.