Directed
Graphs
Copyright © Albert R. Meyer, 2007. All rights reserved.
October 17, 2007
lec 7W.1
A directed graph or digraph:
Definition:
A digraph D is a pair (V,E)
V a set and E a subset of VxV
Elements of V called vertices
Elements of E called directed edges
Ex:
D=(V,E) where
V={a,b,c,d,e,f}
E={(a,a),(a,e),(b,c),(b,e),(b,f),(c,f),(f,d),(f,e)}
Copyright © Albert R. Meyer, 2007. All rights reserved.
October 17, 2007
lec 7W.2
A digraph D:
D=(V,E)
V={a,b,c,d,e,f}
E={(a,a),(a,e),(b,c),(b,e),(b,f),(c,f),(f,d),(f,e)}
and a drawing of D:
d
b
c
a
f
e
Copyright © Albert R. Meyer, 2007. All rights reserved.
October 17, 2007
lec 7W.3
Digraphs
a set, V, of vertices
a set, E  VV
of directed edges.
(v,w)  E notation: vw
Copyright © Albert R. Meyer, 2007. All rights reserved.
October 17, 2007
lec 7W.4
Digraphs
Formally, a Digraph, D, is
exactly the same as a binary
relation on the vertices.
ir reflexive:
asymmetric:
Copyright © Albert R. Meyer, 2007. All rights reserved.
October 17, 2007
lec 7W.5
Digraphs
paths are directed:
v0, v1,…, vn
where vivi+1 for all i
v0
v1
Copyright © Albert R. Meyer, 2007. All rights reserved.
v2
…
October 17, 2007
vn-1
vn
lec 7W.6
Directed Cycles
v0
v1
v2
…
vn-1
v0
vi
vi+1
v0
Copyright © Albert R. Meyer, 2007. All rights reserved.
October 17, 2007
lec 7W.7
Path Relation: Connectedness
v is connected to w:
there is a path
vw
(length 0 path from v to v)
Copyright © Albert R. Meyer, 2007. All rights reserved.
October 17, 2007
lec 7W.8
v connected to w:
a connected to a and e
b connected to b,c,d,e, and f
d connected to d
Etc.
d
b
c
a
f
e
Copyright © Albert R. Meyer, 2007. All rights reserved.
October 17, 2007
lec 7W.9
Positive Path Relation
v is connected to w by a
positive length path
Copyright © Albert R. Meyer, 2007. All rights reserved.
October 17, 2007
lec 7W.10
Connected, pos path length:
a has pos length path to a and e
b has pos length path to c,d,e, and f (not b)
d has pos length path to nothing (not d)
Etc.
d
b
c
a
f
e
Copyright © Albert R. Meyer, 2007. All rights reserved.
October 17, 2007
lec 7W.11
Directed Acyclic Graph’s
DAG’s represent strict partial orders:
•The positive path relation of a DAG
is a strict p.o.
•Every partial order is the positive path
relation of a DAG.
Copyright © Albert R. Meyer, 2007. All rights reserved.
October 17, 2007
lec 7W.12