Combinatorial Mathematics by Holton D.A. (ed.)

By Holton D.A. (ed.)

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G/ D A [ B, and the closed walk v1 ; e1 ; v2 ; : : : ; vk ; ek ; vkC1 defines some circuit in G. g. v1 2 A. But then v2 2 B, v3 2 A, and so on. We conclude that vi 2 A if and only if i is odd. But vkC1 D v1 2 A, so k must be even. 17). 18). Let T be the resulting BFS-tree. G/ n A. If there is an edge e D fx; yg in GŒA or GŒB, the x-y-path in T together with e forms an odd circuit in G. If there is no such edge, we have a bipartition. 5 Planarity We often draw graphs in the plane. A graph is called planar if it can be drawn such that no pair of edges intersect.

Both BFS and DFS occur as subroutines in many other combinatorial algorithms. Some examples will appear in later chapters. Sometimes one is interested in higher connectivity. Let k 2. An undirected graph with more than k vertices and the property that it remains connected even if we delete any k 1 vertices, is called k-connected. A graph with at least two vertices is k-edge-connected if it remains connected after deleting any k 1 edges. So a connected graph with at least three vertices is 2-connected (2-edge-connected) if and only if it has no articulation vertex (no bridge, respectively).

Hint: Group the strings according to the first bit and sort each group. 8. e. an C 1// time. Hint: First sort the strings encoding the numbers according to their length. Then apply the algorithm of Exercise 7. Note: The algorithm discussed in this and the previous exercise is often called radix sorting. , and Stein, C. [2001]: Introduction to Algorithms. Second Edition. E. [1968]: The Art of Computer Programming; Vol. 1. Fundamental Algorithms. D. [1974]: The Design and Analysis of Computer Algorithms.

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