Discrrete mathematics for computer science digraphs

Lemma: The shortest walk between two vertices is a path!. Proof: by contradiction suppose path from u to v crossed itself: c Walks & Paths... Proof: by contradiction suppose path from u

Directed Graphs

Normal Person’s Graph

y

y = f(x)

Computer Scientist’s Graph

a

f

e

c

d

b

• a set, V, of vertices

aka “nodes”

• a set, E ⊆ V×V

of directed edges

(v,w) ∈ E

notation: v→w

Relations and Graphs

a

c

b

d

V= {a,b,c,d}

E = {(a,b), (a,c), (c,b)}

Formally, a digraph with vertices

V is the same as a binary relation

on V

Digraphs

Walks & Paths

Walk : follow successive edges

length: 5 edges

( not the 6 vertices )

Walks & Paths

Path : walk thru vertices without repeat vertex

length: 4 edges

Lemma:

The shortest walk between two vertices is a path!

Proof: (by contradiction) suppose

path from u to v crossed itself:

c Walks & Paths

Proof: (by contradiction) suppose path from u to v

crossed itself:

then path without c -c is

shorter !

Lemma:

The shortest walk between two vertices is a path!

Walks & Paths

Walks & Paths

Digraph G defines walk

relation G+

u G+ v iff ∃walk u to v

(the positive walk relation)

“+” means 1 or more

Walks & Paths

Digraph G defines walk

relation G*

u G* v iff u to v∃ walk

1 2 3

A cycle is a walk whose

only repeat vertex is its

start & end

(a single vertex is a

length 0 cycle)

v0

v0

vi

vi+1

Cycles

Closed walk starts & ends at the

same vertex.

Lemma: The shortest positive

length closed walk containing a

vertex is a positive length cycle!

Proof: similar

Closed Walks & Cycles

has no positive length cycle

D irected A cyclic G raph

DAG

< relation on integers

⊊ relation on sets

prerequisite on classes

D irected A cyclic G raph

DAG

Example: Tournament Graph

• Every team plays every other

H