Commodity - Graph Theory - Home Work, Exercises of Discrete Structures and Graph Theory

During the course work of the Graph Theory, we study the key concept about the:Commodity, Shipped, Maximum, Shipped, Directed Graph, Edges, Xed Limit, Maximum Capacity, Vertex-Capacitated Graph, Matching

Typology: Exercises

2012/2013

Uploaded on 04/29/2013

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HW 8: due Thursday, November 8 in class
1. A certain commodity is produced at two factories x1and x2. The com-
modity is to be shipped to markets y1,y2and y3through the network
shown below. What is the maximum ammount that can be shipped
from the factories to the markets?
x1
x2
ab
cd
y1
y2
y3
5
7
18 4
2
7
12
15
4
6
24
e
24
22
19
3713
16
8
2
2. Let Gbe a directed graph with source sand sink t. Suppose the
capacities are specified not on the edges of Gbut on the vertices (other
than s,t); for each vertex there is a fixed limit on the total flow through
it. There is no restriction on flow through the edges. Show how to use
the ordinary network flow theory to determine the maximum capacity
of a feasible flow from sto tin the vertex-capacitated graph G.
3. Use network flows to prove the oning-Egev´ary theorem, i.e. if Gis
bipartite, then the size of the maximal matching is equal to the size of
the minimum vertex cover.
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