Exercises 7

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					4 – Tell whether the weighted directed graphs are a transportation network or not. If so,
identify the source and sink. If not, tell why.

10 - In this transportation network, the first number along each arc givens the capacity of the
arc. Tell whether the second set of numbers along the arcs is a flow for the network. If so, give
the value of the flow. If not, tell why.

16 - For the network in the previous problem tell whether the given sets S, T forms a cut for the
network indicated. If so, give the capacity of the cut. If not, tell why. S = {A, B, C, D} and T
= {D, E, F}. Also find by inspection:
      a flow value of 17
      a cut of capacity 17

32 – A power generator at a dam can send:
      300 megawatts to substation 1
      200 megawatts to substation 2
      250 megawatts to substation 3.
 In addition…
      substation 2 can send 100 megawatts to substation 1 and 70 megawatts to substation 3
      substation 1 can send at most 280 megawatts to the distribution center
      substation 3 can send at most 300 megawatts to the distribution center.
 Draw a transportation network displaying this information.

4 – Determine the amount by which the flow can be increased along the given path of the
following network, flow, flow augmenting path.
Path = D, B, C, E, F

6 – By performing the ‘flow augmentation algorithm’ on this network and flow, we obtain the
labels shown. Determine a flow having a larger value than the given flow by performing steps
3.1 and 3.2 of the flow augmentation algorithm.

12 – Use the ‘flow augmentation algorithm’ on the following network and flow to show that the
given flow is maximal or else find a flow with a larger value. If the given flow is not maximal,
name the flow-augmenting path and the amount by which the flow can be increased.

18 – Use the flow augmentation algorithm to find a maximal flow for the following
transportation network and flow.
24 – Fins a maximal flow in the following transportation networks by starting with the flow that
is 0 along every arc and applying the ‘flow augmentation algorithm’


8 – Find a maximal cut for the given network (with maximal flow) by applying the ‘flow
augmentation algorithm’.
22 – Consider an undirected graph G in which edge {X, Y} is assigned a nonnegative number c(X,
Y) = c(Y, X) representing its capacity to transmit the flow of some substance in either direction.
Suppose that we want to find the maximum possible flow between distinct verticies S and T of
G, subject to the condition that, for any vertex X (other than S and T) the total flow into X must
equal the total flow out of X. This problem can be solved with the flow augmentation algorithm
by replacing each edge {X, Y} of G by two directed edges (X, Y) and (Y, X) each having a
capacity c(X, Y). Using this method find the maximal and possible flow from S to T if the
numbers on the edges represent the capacity of flow along the edge in either direction


6 – Determine whether the given graph is bipartite or not. If it is, construct the network
associated with the graph.

8 – The bipartite graph is given with a matching given in color. Construct a network associated
with the given graph and use the ‘flow augmentation algorithm’ to determine whether this is a
maximum matching. If not, find the larger matching.

12 – Use the ‘flow augmentation algorithm’ to find a maximum matching for the given bipartite
16 – Five actresses are needed for parts in a ply that require Chinese, Danish, English, French
and German accents.
     Sally does English and French.
     Tess does Chinese, Danish and German.
     Ursula does English and French.
     Vickie does all accents except English.
     Winona does all except Danish and German.
Can the five roles be filled under these conditions? If so, how?

22 – Five men and five women are attending a dance.
      Ann will dance only with Gregory or Harry
      Betty will dance only with Frank or Ian
      Carol will dance only with Harry or Jim
      Diane will dance only with Frank or Gregory
      Ellen will dance only with Gregory or Ian
Is it possible for all 10 people to dance the last dance with an acceptable partner? If so, how?

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