1. Directed Graphs 3/6/121

2. Normal Person’s Graph x y y = f(x) 3/6/122

3. Computer Scientist’s Graph a f e c d b 3/6/123

4. Digraphs a set, V, of vertices aka “nodes” a set, E ⊆ V×V of directed edges (v,w) ∈ E v w notation: v → w 3/6/124

5. Relations and Graphs a c b d V= {a,b,c,d} E = {(a,b), (a,c), (c,b)} 3/6/125

6. Formally, a digraph with vertices V is the same as a binary relation on V. Digraphs 3/6/126

7. Walks & Paths Walk: follow successive edges length: 5 edges (not the 6 vertices) 3/6/127

8. Walks & Paths Path: walk thru vertices without repeat vertex length: 4 edges 3/6/128

9. Lemma: The shortest walk between two vertices is a path! Proof: (by contradiction ) suppose path from u to v crossed itself: u v c Walks & Paths 3/6/129

10. Proof: (by contradiction ) suppose path from u to v crossed itself: u v c then path without c---c is shorter! Lemma: The shortest walk between two vertices is a path! Walks & Paths 3/6/1210

11. Walks & Paths Digraph G defines walk relation G + u G + v iff ∃ walk u to v (the positive walk relation) “+” means 1 or more 3/6/1211

12. Walks & Paths Digraph G defines walk relation G * u G * v iff u to v (the walk relation) “*” means “0 or more” 3/6/1212

13. Cycles A cycle is a walk whose only repeat vertex is its start & end. (a single vertex is a length 0 cycle) 3/6/1213

14. … v0v0 v1v1 v2v2 v n-1 v0v0 v0v0 vivi v i+1 Cycles 3/6/1214

15. 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 3/6/1215

16. has no positive length cycle Directed Acyclic Graph DAG lec 7M.16 3/6/1216

17. examples: < relation on integers ⊊ relation on sets prerequisite on classes Directed Acyclic Graph DAG 3/6/1217

18. Example: Tournament Graph Every team plays every other 3/6/1218 H Y P D H Y P D DAG => Unique ranking