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Description Of A Graph

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Use the map to create a graph where vertices represent street intersections and edges represent streets. Define c(u,v) = 1 for all edges (u,v). Since a street can be traversed, start off by creating a directed edge in each direction, then make the transformation to a flow problem with no antiparallel edges as described in the section. Make the home the source and the school the sink. If there exist at least two distinct paths from source to sink then the flow will be at least 2 because we could assign f(u,v) = 1 for each of those edges. However, if there is at most one distinct path from source to sink then there must exist a bridge edge (u, v) whose removal would disconnect s from t. Since c(u, v) = 1, the flow into u is at most 1. We may …show more content…

Exercise 26.2-10
Suppose we already have a maximum flow f. Consider a new graph G where we set the capacity of edge (u, v) to f (u, v). Run Ford-Fulkerson, with the mod- ification that we remove an edge if its flow reaches its capacity. In other words, if f(u,v) = c(u,v) then there should be no reverse edge appearing in residual network. This will still produce correct output in our case because we never exceed the actual maximum flow through an edge, so it is never advantageous to cancel flow. The augmenting paths chosen in this modified version of Ford- Fulkerson are precisely the ones we want. There are at most |E| because every augmenting path produces at least one edge whose flow is equal to its capacity, which we set to be the actual flow for the edge in a maximum flow, and our modification prevents us from ever destroying this progress.

Problem 26-5
a. Since the capacity of a cut is the sum of the capacity of the edges going from a vertex on one side to a vertex on the other, it is less than or equal to the sum of the capacities of all of the edges. Since each of the edges has a capacity that is ≤ C, if we were to replace the capacity of each edge with C, we would only be potentially increasing the sum of the capacities of all the edges. After so changing the capacities of the edges, the sum of the capacities of all the edges is equal to C|E|, potentially an overestimate of the original capacity of any cut, and so of the minimum cut.
b.

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