ENGM 732 Formalization of Network Flows Network Flow Models

Terminology Loop 5 Graph

Terminology Graph Multigraph

Terminology Directed Network Graph

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

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

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

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

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

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

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

Expanded Network Directed Network Expanded Network o(-k) = t(k) t(-k) = o(k)

Expanded Network Marginal Network Consists of a subset that are admissible arcs (2,2,-1) (f k, c k, h k ) Network with flow