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 Vertex v has outdegree 3 Vertex has indegree 2 v

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

5. Tournament Graph 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 iff it is a DAG. H Y P D H Y P D

6. Tournament Graphs and Rankings Theorem: A tournament graph determines a unique ranking iff it 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, v 1, …, v n, such that V = {v 1, … v n } and for any i and j, i<j implies v i →v j.

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 v 1, … v n. Then v, v 1, … v n is a unique ranking for the vertices of G. Vertex v can only go at the beginning of the list since v→v i for i = 1, … n (why?).

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