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By Timothy A. Davis
This books presents an honest library of sparse matrix capabilities. notwithstanding, it may be obscure the code from time to time as the writer selected to take advantage of cryptic variable names.
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Example text
K(χ(G) − 1) + 1. Now let us show that the span is at least the right hand side. Let t be the span, and consider a feasible assignment φ using channels 0, 1, . . , t − 1 which uses as few as possible channels which are not multiples of k. Then in fact φ must use only multiples of k, for otherwise the least channel not a multiple of k could be pushed down to the nearest multiple of k, giving a contradiction. But now if we let c(v) = φ(v)/k we obtain a (proper) 30 McDiarmid colouring of G, and so χ(G) ≤ (t − 1)/k + 1, which yields the desired inequality.
4. 1. 1 21 Optimal Ear Decompositions of Matching Covered Graphs Let G be a matching covered graph. If G is a brick or a brace, we have seen how to find optimal ear decompositions of G. Suppose that G is neither a brick nor a brace. Then it has nontrivial tight cuts. Unfortunately, there is no obvious way of obtaining an ear decomposition (much less an optimal ear decomposition) of G from arbitrary (optimal) ear decompositions of G1 and G2 , where G1 and G2 are the two C-contractions of G with respect to a tight cut C of G.
2 Bounded tree-width graphs The ‘tree-width’ of a connected graph measures how far the graph is from being a tree – see for example Chapter 4 in this book. On trees, many problems can be solved quickly (in polynomial time) by simple dynamic 40 McDiarmid programming, and often a similar approach works for graphs of bounded tree-width. For example it is easy to determine the chromatic number of such graphs. 5). For such problems, if we consider graphs of bounded tree-width, the standard dynamic programming approach will determine the span in polynomial time.