Discrete and Computational Geometry: Japanese Conference, by Jin Akiyama, Mikio Kano

By Jin Akiyama, Mikio Kano

This publication constitutes the completely refereed post-proceedings of the japanese convention on Discrete Computational Geometry, JCDCG 2002, held in Tokyo, Japan, in December 2002.

The 29 revised complete papers offered have been rigorously chosen in the course of rounds of reviewing and development. All present concerns in discrete algorithmic geometry are addressed.

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Extra resources for Discrete and Computational Geometry: Japanese Conference, JCDCG 2002, Tokyo, Japan, December 6-9, 2002. Revised Papers

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Thus, exactly those vertices of GT corresponding to inner triangles have degree three, whereas all other vertices have degree one (ears of the triangulation) or two. 7 (a) (b) 6 6 5 4 (c) Fig. 1. Different incarnations of the triangulation coloring game Motivation for considering the dual graph of T (S) stems from the following observation. Coloring an edge of a triangle ∆ for which one edge has already been colored leads immediately to a winning move for the opponent: she just has to color the third edge of ∆.

When the player can win, a winning sequence of moves can be found within the same time bound. Proof. Let S be the triangulated subpolygon to the right of a given oriented diagonal d. There are two diagonals d1 and d2 in S, that form a triangle together with d, which we orient leaving d to their left, as shown in Figure 3. Let us denote by S 1 and S 2 the subpolygons these diagonals define (we follow the counterclockwise order). When the notation is iterated we write simply S i,j instead of (S i )j .

The player of the Green-Wins Solitaire Game can always win for any given triangulation on n ≥ 4 points in convex position. Proof. The number of edges in any triangulation is 2n − 3, therefore we have to prove that we can always achieve at least n − 1 green edges after a suitable sequence of flips. We proceed by induction. The cases n = 4, 5, 6 are easily checked directly, hence we can assume n ≥ 7. Let a and b be consecutive boundary edges of an ear of the triangulation, and let d be the diagonal which completes a triangle with a and b (refer to Figure 6).

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