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35 TREE OF A GRAPH directed segment, a link a nondirected segment. I t is assumed that the scale of the two coordinate axes is the same, so that the idea of an angle acquires meaning, which it obviously does not possess in a graph drawn in accordance with the theory of sets; that is, in the sense given to it by Konig and Berge. Sollin’s method is obviously equally suitable for a nontopological graph drawn on a map. Final Optimization in the Case of a Metric Graph. If it is decided that a network of pipes must be a tree, there is no a priori way of knowing the number of vertices required in an actual case.

Author’s observation. 6. SEARCH FOR AN OPTIMAL FLOW I N A NETWORK 43 Each overseas connection between a port Ai and a port Bj has a limited capacitycij . T h e problem is to organize the shipments so as to fulfill all the orders in the best possible way. For this purpose, a transport network is constructed by connecting a point of entry A to the m vertices A i by arcs with a capacity x i , and the n vertices Bj to an exit B by arcs with a capacity yj . , m. j=1 It should be added that certain capacities cij may be null on the hypothesis that there is no overseas connection in their case.

If either of the points Y , or Y , is a hanging vertex it must be rejected for the same reason. If neither of the vertices is hanging it necessarily follows, since the tree has four links, that a chain exists of the form YIY,X or Y2Y,X, where X may be either A , B, or C. If these vertices are aligned, this means that one of the two Y’s is not to be considered. If these vertices are not aligned, the broken line coinciding with the chain is longer than the straight line coinciding with link Y,X or Y z X ,which equally leads to the rejection of one of the Y’s.

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