network-flow


Arbitrary flow network with source sink and positive capacity on every edge


let G = (V,E) be an arbitrary flow network, with a source s, a sink t, and a positive integer capacity Ce on every edge e.
(a) Assuming there is at least one s-t path in G. If all edge capacities of edges in E are different, then there is only one min-cut (A,B) for G.
(b) Define G'= (V,E') to be a flow network with the same vertices V as G, and the edges E' consist of the same edges in E, except that the capacities of edges in E' are twice the capacity of the corresponding edge in E. That is, e = (u,v)∈ E has capacity Ce if and only if e' = (u,v) ∈ E' has capacity 2Ce. Then the capacity of any min-cut of G' has twice the capacity of any min-cut of G.
Need some help with this question, thanks.
please state if each claim is true of false, need explanation or counter-example, Thank you.

Related Links

Network flow: the number of optimal augmenting paths
Arbitrary flow network with source sink and positive capacity on every edge
Maximum flow in the undirected graph

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