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That means that the vertices of a bipartite graph can be divided into two classes ‘R’ and ‘Y’ such that no edge of the graph runs between two ‘R’ vertices or between two ‘Y’ vertices. Bipartite graphs are most often drawn, as in Fig. 5, in two layers, with all edges running between layers. Fig. 5: A bipartite graph The complement G of a graph G is the graph that has the same vertex set that G has and has an edge exactly where G does not have its edges. Formally, E(G) = {(v, w) | v, w ∈ V (G); v = w; (v, w) ∈ / E(G)}.

N. One difference that this makes is that there are a lot more labeled graphs than there are unlabeled graphs. There are, for example, 3 labeled graphs that have 3 vertices and 1 edge. They are shown in Fig. 7. Fig. 7: Three labeled graphs... There is, however, only 1 unlabeled graph that has 3 vertices and 1 edge, as shown in Fig. 8. Fig. 8: ... but only one unlabeled graph Most counting problems on graphs are much easier for labeled than for unlabeled graphs. Consider the following question: how many graphs are there that have exactly n vertices?

Hence, L(n) = (1 + 2 + · · · + (n − 1)) = Θ(n2 ). The worst-case behavior of Quicksort is therefore quadratic in n. In its worst moods, therefore, it is as bad as ‘slowsort’ above. Whereas the performance of slowsort is pretty much always quadratic, no matter what the input is, Quicksort is usually a lot faster than its worst case discussed above. We want to show that on the average the running time of Quicksort is O(n log n). The first step is to get quite clear about what the word ‘average’ refers to.