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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 graph. 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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posted: | 10/23/2011 |

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