# Graph Theory in Operations Research by T. B. Boffey (auth.) PDF

By T. B. Boffey (auth.)

ISBN-10: 0333282140

ISBN-13: 9780333282144

ISBN-10: 1349166758

ISBN-13: 9781349166756

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Example text

However, in practice r(X) is taken to be a function only of the level of X (that is, p). 1 and r(X) = n - p is sometimes used. 3a shows a complete enumeration tree, for n=4 and A= {a, b, c, d} generated by rank branching. The subset of solutions corresponding to a particular node will depend on the selection rule used! For example, consider the nodes marked U, V, Wand Y at levels 0, 1, 2 and 3 respectively. Then Y corresponds to the single solution (b, (b, (a, (d, d, a, c) a, d, c) b, c, d) c, b, a) r(U) = 3, r(V) = 1 and r(W) = 4 r(U) = 2, r(V) = 1 and r(W) = 4 r(X) = (level of X) + 1 for all X r(X) = n- (level of X) for all X if if if if (2) Immediate successor branching A node X is specified by p (=level of X) relations of the form rj = ri + 1 (to be interpreted as item aj is ranked immediately after item ai).

This problem is equivalent to the celebrated Konigsberg bridges problem which gave rise to what is often quoted as the earliest paper on graphs (by Euler in 1736). 14) once and once only. 15) both of which meet the requirements regarding display areas and shapes, demand for utilities, etc. It is intended that all visitors should follow a fixed route whl~h passes along each line of displays just once (that is, once in each direction along avenues with displays on both sides). Moreover, it is also desirable that the stream of visitors should not 'cross itself.

But u 1 cannot belong to both X andY so the premise that a cycle with an odd number of edges exists must be false. D Corollary edges. 3 vertices, whose cycles all have at least four edges. For G to be planar q~2n-4 must hold. 3 and is left as an exercise. Corollary q~2n-4 For a bipartite undirected graph G on n;;;;o3 vertices to be planar, must hold. 3 planar. 5 does not establish the non-planarity of the graph. However, removing 5-6 and contracting 4-5 to nothing (and coalescing its end vertices) results in a graph which again has cycles each with at least four edges.