## Flow Networks and FlowsFlow Network is a directed graph that is used for modeling material Flow. There are two different vertices; one is a Some real-life problems like the flow of liquids through pipes, the current through wires and delivery of goods can be modeled using flow networks.
- For each edge (u, v) ∈ E, we associate a nonnegative weight capacity c (u, v) ≥ 0.If (u, v) ∉ E, we assume that c (u, v) = 0.
- There are two distinguishing points, the source s, and the sink t;
- For every vertex v ∈ V, there is a path from s to t containing v.
Let G = (V, E) be a flow network. Let s be the source of the network, and let t be the sink. A flow in G is a real-valued function f: V x V→R such that the following properties hold: **Capacity Constraint:**For all u, v ∈ V, we need f (u, v) ≤ c (u, v).**Skew Symmetry:**For all u, v ∈ V, we need f (u, v) = - f (u, v).**Flow Conservation:**For all u ∈ V-{s, t}, we need
The quantity f (u, v), which can be positive or negative, is known as the net flow from vertex u to vertex v. In the The three properties can be described as follows: **Capacity Constraint**makes sure that the flow through each edge is not greater than the capacity.**Skew Symmetry**means that the flow from u to v is the negative of the flow from v to u.- The flow-conservation property says that the total net flow out of a vertex other than the source or sink is 0. In other words, the amount of flow into a v is the same as the amount of flow out of v for every vertex v ∈ V - {s, t}
The value of the flow is the net flow from the source, The The A flow f is said to be Next TopicNetwork Flow Problems |