Residual Graph Edge u v flow value fe

Residual Graph Edge: • u v • flow value fe, • Capacity ce Residual graph Gf • ce – fe in u v direction • fe in v u direction

My preferred notations fe/ce u v c e - fe v u fe

Unit capacity graphs v unused in f u v v u

Example 7. 8 from textbook G s f s a b c d u v w x t Gf s a b c d s t path in Gf a t s u v w x t u t

Add path, repeat G s f s a b c d u v w x t Gf s a b c d s t path in Gf a t s c u v w x u v t t

Adding flows f s a b c d s t path in Gf u v w x a t s s a b c d u v w x c t u v t

Adding flows f s a b c d s t path in Gf u v w x a t s c cancellation s a b c d u v w x t u v t

Add path, repeat G s f s a b c d u v w x t Gf s a b c d u v w x t s t path in Gf t s b w t

Add path, repeat !? !? !@#$ G s a b c d u v w x t f Gf s a b c d u v w x t

No path in Gf G s f s a b c d u v w x t Gf s a b c d u v w x t Reachable from S

Cut {S, b, d, w} G s a b c d u v w x f t s a b c d u v w x t Flow leaving: 3

Cut {S, b, d, w} G s a b c d u v w x Capacity leaving: 3 f t s a b c d u v w x t

Matching interpretation f s a b c d u v w x t a b c u v w d x

Bipartite view: left side set: {b, d} G s a b c d u v w x Capacity leaving: 3 t b w d Certificate against perfect matching