Tournament (graph theory)

Tournament
A tournament on 4 vertices
Vertices	${\displaystyle n}$
Edges	${\displaystyle {\binom {n}{2}}}$

A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph. That is, it is an orientation of a complete graph, or equivalently a directed graph in which every pair of distinct vertices is connected by a single directed edge.

Many of the important properties of tournaments were first investigated by Landau (1953) in order to model dominance relations in flocks of chickens. Current applications of tournaments include the study of voting theory and social choice theory among other things. The name tournament originates from such a graph's interpretation as the outcome of a round-robin tournament in which every player encounters every other player exactly once, and in which no draws occur. In the tournament digraph, the vertices correspond to the players. The edge between each pair of players is oriented from the winner to the loser. If player ${\displaystyle a}$ beats player ${\displaystyle b}$, then it is said that ${\displaystyle a}$ dominates ${\displaystyle b}$.