Discrrete mathematics for computer science DAG warmup

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

DAG Warm-Up Problem

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.

Indegree and outdegree of a vertex in a digraph

• Vertex v has outdegree 3

• Vertex has indegree 2

v

• Lemma Any finite DAG has at least one node of indegree 0.

• Proof In-class exercise

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

Tournament Graphs and Rankings

• Theorem: A tournament graph determines a unique ranking iff it

is a DAG

• What does this mean?

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

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→vi for i = 1, … n (why?)

If a tournament graph G determines a unique ranking,

then G is a DAG

• Proof: Exercise