Tournament (graph theory)

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 directed edge (often, called an arc) with any one of the two possible orientations.

Tournament
A tournament on 4 vertices
Vertices	${\displaystyle n}$
Edges	${\displaystyle {\binom {n}{2}}}$
Table of graphs and parameters

Many of the important properties of tournaments were first investigated by H. G. Landau in Landau (1953) 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 ${\displaystyle b}$. If every player beats the same number of other players (indegree = outdegree), the tournament is called regular.