Directed Graphs
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Normal Person’s Graph
y
y = f(x)
x
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Computer Scientist’s Graph
a
b
c
d
f
e
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Digraphs
• a set, V, of vertices
aka “nodes”
• a set, E  V×V
of directed edges
(v,w)  E notation: vw
v
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w
4
Relations and Graphs
a
d
b
c
V= {a,b,c,d}
E = {(a,b), (a,c), (c,b)}
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Digraphs
Formally, a digraph
with vertices V is the
same as a binary
relation on V.
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Walks & Paths
Walk: follow successive edges
length: 5 edges
(not the 6 vertices)
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Walks & Paths
Path: walk thru vertices
without repeat vertex
length: 4 edges
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Walks & Paths
Lemma:
The shortest walk between
two vertices is a path!
Proof: (by contradiction) suppose
path from u to v crossed itself:
c
u
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v
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Walks & Paths
Lemma:
The shortest walk between
two vertices is a path!
Proof:
(by contradiction)
then path
without c---csuppose
is
path
from u to v crossed itself:
shorter!
c
u
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v
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Walks & Paths
Digraph G defines walk
+
relation G
+
u G v iff ∃walk u to v
(the positive walk relation)
“+” means 1 or more
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Walks & Paths
Digraph G defines walk
*
relation G
*
u G v iff 1 w
u
to
v
alk
2 3
length  0
(the walk relation)
“*” means “0 or more”
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Cycles
A cycle is a walk whose
only repeat vertex is its
start & end.
(a single vertex is a
length 0 cycle)
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Cycles
…
v0
v1
v2
vn-1
v0
vi
v0
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vi+1
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Closed Walks & 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
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Directed Acyclic Graph
DAG
has no positive
length cycle
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lec 7M.16
Directed Acyclic Graph
examples:
DAG
< relation on integers
⊊ relation on sets
prerequisite on classes
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Example: Tournament Graph
• Every team plays every other
H
H
Y
P
D
Y
P
D
DAG => Unique ranking
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