1. ENGM 732 Formalization of Network Flows Network Flow Models

2. Terminology 1 4 3 2 1 Loop 5 Graph

3. Terminology 1 4 3 2 5 Graph 1 4 3 2 5 Multigraph

4. Terminology 1 4 3 2 5 Directed Network 1 4 3 2 5 Graph

5. Terminology Walk – alternating sequence of nodes and lines beginning and ending with nodes Closed walk – same node begins and ends walk Open walk – ends at node other than beginning node Trail – walk with no line repeated Path - a walk with no node repeated

6. Terminology Walk – alternating sequence of nodes and lines beginning and ending with nodes Open walk – ends at node other than beginning node 45

7. Terminology Closed walk – same node begins and ends walk Trail – walk with no line repeated 1 4 2

8. Terminology Path - a walk with no node repeated 1 4 2

9. Terminology - Trees [ b i ] (c k, h k ) [3] [-5] (1,2) (2,-1) (3,5) 2 1 3 4 (2,1) (5,3) [0] 1 2 5 4 3 5 7 6 (1,-1) (2,1) 5 Root = 5 N T = {5} M T = 

10. Terminology - Trees [ b i ] (c k, h k ) [3] [-5] (1,2) (2,-1) (3,5) 2 1 3 4 (2,1) (5,3) [0] 1 2 5 4 3 5 7 6 (1,-1) (2,1) Root = 5 N T = {1,2,3,4,5} M T = {1,2,5,7} 2 1 3 4 1 2 5 7 5 Spanning Tree

11. Terminology - Trees [ b i ] (c k, h k ) [3] [-5] (1,2) (2,-1) (3,5) 2 1 3 4 (2,1) (5,3) [0] 1 2 5 4 3 5 7 6 (1,-1) (2,1) Root = 5 N T = {1,2, 5} M T = {1,7} 2 1 3 4 1 5 7 5 Spanning Forest Root = 3 N T = {3, 4} M T = {5}

12. Expanded Network 2 1 3 4 1 2 5 4 3 5 8 7 6 2 1 3 4 1 2 5 4 3 5 8 7 6 Directed Network Expanded Network o(-k) = t(k) t(-k) = o(k) -6 -7 -2 -8 -3 -4 -5

13. Expanded Network 2 1 3 4 1 2 5 4 3 5 8 7 6 2 1 3 4 1 2 5 4 3 5 8 7 Marginal Network Consists of a subset that are admissible arcs -6 -7 -2 -8 -3 -4 -5 (2,2,-1) (f k, c k, h k ) Network with flow