Graph-Theoretic Concepts in Computer Science: 32nd by Fedor V. Fomin

By Fedor V. Fomin

This e-book constitutes the completely refereed post-proceedings of the thirty second overseas Workshop on Graph-Theoretic techniques in desktop technological know-how, WG 2006, held in Bergen, Norway in June 2006. The 30 revised complete papers awarded including one invited paper have been conscientiously chosen from ninety one submissions. The papers handle all facets of graph-theoretic recommendations in desktop science.

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The following two statements are equivalent: 1. G(V, E) is chordal. 2. , Tu and Tv have a common node). For clarity, we will use the word “vertex” when we refer to the graph G(V, E), and “node” when referring to T (U, F ). The tree T together with the subtrees Tv is called the tree decomposition of G. A tree decomposition of G can be found in polynomial time (see [4,9]). We say that a vertex v contains node x if Tv contains node x. Similarly, node x can be reached from vertex v is shorthand for saying that there is a vertex v that contains x and there is a path between v and v .

4 Clique Reduction Henceforth it is assumed that W = {w1 , w2 , . . , wk+1 } is a hole cover of G. As in the previous section, let V0 = V \ W and denote by G0 the chordal graph G \ W . In this section we show that if there is a large clique K in G0 , then K contains a irrelevant vertex. 42 D. Marx In the rest of the paper, we prove several lemmas that state certain properties of the instance. However, these properties do not always hold, but in this case the compression algorithm can identify and return a necessary set.

Repeat this until every path is either selected or thrown away. In each step we select one path and throw away less than k 4 paths, thus |A| > k 6 implies that we can select more than k 2 paths. Thus assume without loss of generality that the paths P1 , . . , Pk2 +1 do not intersect each other outside the clique K. If a vertex u ∈ K is contained in more than k + 1 of these paths, then u is a necessary vertex. To see this, notice that for each i = 1, . . , k + 1 a W -avoiding size k hole cover has to contain at least one vertex of each Pi (Prop.

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