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where (G, 0) = 1 and (G, 1) is the number of edges in G Let us deviate for a moment and de ne the graphical invariant, k-factor To do so, we rst de ne a few other graph theoretic terms A graph is k-regular if every vertex has degree k A graph H is a spanning subgraph of G if it is a subgraph that has the same vertex set of G A subgraph H is a k-factor if it is a k-regular spanning subgraph A 1-factor is a spanning set of edges and a 2-factor of a graph G is a collection of cycle subgraphs that span the vertex set of G If the collection of spanning cycles consists of a single cycle, then the graph is Hamiltonian Hamiltonian theory is an area that has received substantial attention among graph theorists, as well as the topic of k-factors.

Financial (banking, insurance, other financial services) 88.5% 61.5 46.2 46.2 53.8 38.5 23.1 15.4 7.7

Obviously, support(D) is also a database over U. Comparing this definition to Definition 5.4.1, we see that U (t) =

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We note that (G, 1) is the number of edges in G and that (G, n/2) is equivalent to the number of 1-factors in G [9] In the following sections, we de ne selected graphical invariants and topological indices, most of which were utilized in the work by Haynes et al [22,23] Domination numbers of graphs have been utilized extensively in elds such as computer network design and fault tolerant computing The idea of domination is based on sets of vertices that are near (dominate) all the vertices of a graph A set of vertices dominate the vertex set if every vertex in the graph is either in the dominating set or adjacent to at least one vertex in the dominating set The minimum cardinality among all dominating sets of vertices in the graph is the domination number For more information on the domination number of graphs see Haynes [25].

Slice 1 2 3 4 5 6 7 8 9 10 Total Utility 20 36 50 58 64 68 70 70 68 64 Marginal Utility 20 16 14 8 6 4 2 0 2 4

If restrictions are placed on the set of vertices that we may select to be in the dominating set, then we obtain variations on the domination number For example, the independent domination number is the minimum number of nonadjacent vertices that can dominate the graph Consider Figure 35, which contains two trees of order 7, one with independent domination number equal to 3 and the other with independent domination number equal to 2 The vertices in each independent minimum dominating set are labeled {u, w, z} and {u, z}, respectively Domination numbers have been highly studied in.

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The reader (especially one who has not been paying close attention) may, at this point, ask, So what The end result is pretty much the same The answer is: The end result is not necessarily the same First, by filing a declaratory judgment action, the loathsome infringer is able to choose the battlefield (If this doesn t seem important, reread the section of this chapter entitled Jurisdiction and Venue ) Second, the disgusting scalawag is able to choose the timing of the action This may be significant for several reasons A patentee may, for example, have planned, indeed prepared, to take action against a different infringer (remember that infringement is often industry-wide) These plans and preparations may well be totally upset when the patentee is unexpectedly hauled into a different court to face a different infringer.

Tim O Reilly, Open Source Paradigm Shift, tim.oreilly.com, (June 2004), available at http://

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