Discrrete mathematics for computer science DAG warmup

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

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Vertex v has outdegree 3
Vertex has indegree 2

v


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


Tournament Graph

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A digraph is a tournament graph iff every pair of distinct nodes is
connected by an edge in exactly one direction.

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Theorem: A tournament graph determines a unique ranking iff it
is a DAG.

H

P

H

Y

D

Y

P
D


Tournament Graphs and Rankings

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Theorem: A tournament graph determines a unique ranking iff it
is a DAG.

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What does this mean?


Tournament Graphs and Rankings

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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

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

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Proof: Exercise