# graph.maxflow

0th

Percentile

##### Maximum flow in a network

In a graph where each edge has a given flow capacity the maximal flow between two vertices is calculated.

Keywords
graphs
##### Usage
graph.maxflow(graph, source, target, capacity=NULL)
graph.mincut(graph, source=NULL, target=NULL, capacity=NULL,
value.only = TRUE)
##### Arguments
graph
The input graph.
source
The id of the source vertex.
target
The id of the target vertex (sometimes also called sink).
capacity
Vector giving the capacity of the edges. If this is NULL (the default) then the capacity edge attribute is used.
value.only
Logical scalar, if TRUE only the minumum cut value is returned, if FALSE the edges in the cut and a the two (or more) partitions are also returned. This currently only works for undirected graphs.
##### Details

graph.maxflow calculates the maximum flow between two vertices in a weighted (ie. valued) graph. A flow from source to target is an assignment of non-negative real numbers to the edges of the graph, satisfying two properties: (1) for each edge the flow (ie. the assigned number) is not more than the capacity of the edge (the capacity parameter or edge attribute), (2) for every vertex, except the source and the target the incoming flow is the same as the outgoing flow. The value of the flow is the incoming flow of the target vertex. The maximum flow is the flow of maximum value.

graph.mincut calculates the minimum st-cut between two vertices in a graph (if the source and target arguments are given) or the minimum cut of the graph (if both source and target are NULL).

The minimum st-cut between source and target is the minimum total weight of edges needed to remove to eliminate all paths from source to target.

The minimum cut of a graph is the minimum total weight of the edges needed to remove to separate the graph into (at least) two components. (Which is to make the graph not strongly connected in the directed case.) The maximum flow between two vertices in a graph is the same as the minimum st-cut, so graph.maxflow and graph.mincut essentially calculate the same quantity, the only difference is that graph.mincut can be invoked without giving the source and target arguments and then minimum of all possible minimum cuts is calculated.

Starting from version 0.5 igraph can return the edges in the minimum cut for undirected graphs.

##### Value

• A numeric constant, the maximum flow, or the minimum cut, except if value.only=FALSE for graph.mincut. In this case a named list with components:
• valueNumeric scalar, the cut value.
• cutNumeric vector, the edges in the cut.
• partition1The vertices in the first partition after the cut edges are removed. Note that these vertices might be actually in different components (after the cut edges are removed), as the graph may fall apart into more than two components.
• partition2The vertices in the second partition after the cut edges are removed. Note that these vertices might be actually in different components (after the cut edges are removed), as the graph may fall apart into more than two components.

• Maximum flow
• Minimum cut

##### References

A. V. Goldberg and R. E. Tarjan: A New Approach to the Maximum Flow Problem'' Journal of the ACM 35:921-940, 1988.