By C. Berge, D. Ray-Chaudhuri

ISBN-10: 3540068465

ISBN-13: 9783540068464

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9. Fig. 7 Find the Hamiltonian cycle of greatest weight in the graph of Fig. 5. Chapter 4 Trees A fool sees not the same tree that a wise man sees. William Blake We are all familiar with the idea of a family tree. In this chapter, we study trees in general, with special reference to spanning trees in a connected graph and to Cayley's celebrated result on the enumeration of labelled trees. The chapter concludes with some further applications. 9 Properties of trees A forest is a graph that contains no cycles, and a connected forest is a tree.

A disconnecting set in a connected graph G is a set of edges whose removal disconnects G. For example, in the graph of Fig. 3, the sets {e\, e^ £5} and {^3, e^, ej, eg} are both disconnecting sets of G; the disconnected graph left after removal of the second is shown in Fig. 4. Fig. 3 Fig. 4 We further define a cutset to be a disconnecting set, no proper subset of which is a disconnecting set. In the above example, only the second disconnecting set is a cutset. Note that the removal of the edges in a cutset always leaves a graph with exactly two components.

We now apply the greedy algorithm to obtain a lower bound for the solution of the travelling salesman problem. This is useful, since the greedy algorithm is an efficient algorithm, whereas no efficient general algorithms are known for the travelling salesman problem. If we take any Hamiltonian cycle in a weighted complete graph and remove any vertex v, then we obtain a semi-Hamiltonian path, and such a path must be a spanning tree. So any solution of the travelling salesman problem must consist of a spanning tree of this type together with two edges incident to v.

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Hypergraph Seminar by C. Berge, D. Ray-Chaudhuri

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