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Tournament Graphs and Rankings Theorem

Understand the uniqueness of rankings in tournament graphs as they relate to directed acyclic graphs (DAGs) and explore the implications of this relationship in graph theory. Learn to identify unique sequences of nodes in DAGs for ranking determinations.

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Tournament Graphs and Rankings Theorem

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  1. DAG Warm-Up Problem

  2. McNally and fellow guard Brandyn Curry, who combined for 26 second-half points, came up big for Harvard throughout the final frame. After going scoreless in the first half, Curry scored 12 straight points for the Crimson off four three-pointers during a stretch of 3:27, turning a one-point deficit into a seven-point Harvard lead.

  3. Indegree and outdegree of a vertex in a digraph v Vertex v has outdegree 3 Vertex has indegree 2

  4. Lemma. Any finite DAG has at least one node of indegree 0. Proof. In-class exercise.

  5. Tournament Graph H H Y Y P P D D A digraph is a tournament graph iff every pair of distinct nodes is connected by an edge in exactly one direction. Theorem: A tournament graph determines a unique ranking iffit is a DAG.

  6. Tournament Graphs and Rankings Theorem: A tournament graph determines a unique ranking iffit is a DAG. What does this mean?

  7. Tournament Graphs and Rankings Theorem: A tournament graph determines a unique ranking iff it is a DAG. What does this mean? That there is a unique sequence of the nodes, v1, …, vn, such that V = {v1, … vn} and for any i and j, i<j implies vi→vj.

  8. If a tournament graph G is a DAG, then G determines a unique ranking Proof by induction on |V|. The base case |V|=1 is trivial. Induction. Suppose |V|=n+1 and every tournament DAG with ≤n vertices determines a unique ranking. G has a unique vertex v of indegree 0. (Why is there a vertex of indegree 0? Why is it unique?) Let S be the set of all vertices w such that there is an edge v→w. (What vertices in V are actually in S?) The edges between nodes in S comprise a tournament DAG (why?) and hence determine a unique ranking v1, … vn. Then v, v1, … vn is a unique ranking for the vertices of G. Vertex v can only go at the beginning of the list since v→vifor i = 1, … n (why?).

  9. If a tournament graph G determines a unique ranking, then G is a DAG Proof: Exercise

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