Computational Geometry and Graph Theory: International by Jin Akiyama, Midori Kobayashi, Gisaku Nakamura (auth.), Hiro

By Jin Akiyama, Midori Kobayashi, Gisaku Nakamura (auth.), Hiro Ito, Mikio Kano, Naoki Katoh, Yushi Uno (eds.)

This publication constitutes the completely refereed post-conference complaints of the Kyoto convention on Computational Geometry and Graph conception, KyotoCGGT 2007, held in Kyoto, Japan, in June 2007, in honor of Jin Akiyama and Vašek Chvátal, at the celebration in their sixtieth birthdays.

The 19 revised complete papers, provided including five invited papers, have been rigorously chosen in the course of rounds of reviewing and development from greater than 60 talks on the convention. All elements of Computational Geometry and Graph conception are coated, together with tilings, polygons, most unlikely items, coloring of graphs, Hamilton cycles, and components of graphs.

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Extra info for Computational Geometry and Graph Theory: International Conference, KyotoCGGT 2007, Kyoto, Japan, June 11-15, 2007. Revised Selected Papers

Example text

Consider a pyramid lpq, and let f1 , f2 , . . , fk be the interior vertices of SPxl xq . −−−−→ Notice that SPxl xq is a convex chain. Definition 10. If all f1 , f2 , . . , fk are on ∂PU then lpq is an upper pyramid. If all f1 , f2 , . . , fk are on ∂PL then lpq is a lower pyramid. If some of f1 , f2 , . . , fk are on ∂PU and the others are on ∂PL , then lpq is a hybrid pyramid. We can prove that hybrid pyramids do not exist in a simple 3-shortcut-free (s, X, t)-path (see Lemma 3). 3 Properties Lemma 1.

KyotoCGGT 2007, LNCS 4535, pp. 41–55, 2008. c Springer-Verlag Berlin Heidelberg 2008 42 O. Daescu and J. Luo P x1 t t −−→ SPst x2 X s s Fig. 1. A simple s-to-t path that turns on points in X (bold line), a shortest s-to-t path that turns on points in X and is not a simple path (dotted line), and the shortest s-to-t path in P (bold dashed line) form and in the special form studied in this paper, has been introduced in [3]. They have motivated it by its applications to polygon generation, where given a set of points X inside a simple polygon P one wishes to generate simple, or x-monotone, polygons with vertices in X [1, 3, 7].

Since Fv contains at most 2q vertices of degree at least 1 and G − Fv has order (∆ − 1)q + t − 1, there are at least 2(∆ − 1)q + 2(t − 1) edges from G − Fv to the nontrivial components of Fv . Since Fv has at most 2q vertices of degree at > ∆ − 1, it follows that there exists a vertex in Fv of least 1 and 2(∆−1)q+2(t−1) 2q degree at least ∆ + 1. Thus we get a contradiction. By Lemma 2, we have a lower bound on the maximum degree of a given graph in terms of its order and its forest number. In other words, if G is a graph of order n, then ∆(G) ≥ f2n (G) − 1.

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