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Journal for Geometry and Graphics 19 (2015), No. 2, 227--236
Copyright Heldermann Verlag 2015



Random Realization of Polyhedral Graphs as Deltahedra

Naoya Tsuruta
School of Media Science, Tokyo University of Technology, 1404-1 Katakuramachi, Hachioji, Tokyo 192-0982, Japan
tsurutany@stf.teu.ac.jp

Jun Mitani
Graduate School of Systems and Information Engineering, University of Tsukuba, Ibaraki 305-8573, Japan
mitani@cs.tsukuba.ac.jp

Yoshihiro Kanamori
Graduate School of Systems and Information Engineering, University of Tsukuba, Ibaraki 305-8573, Japan
kanamori@cs.tsukuba.ac.jp

Yukio Fukui
Graduate School of Systems and Information Engineering, University of Tsukuba, Ibaraki 305-8573, Japan
fukui@cs.tsukuba.ac.jp



Ee propose a method for realizing a polyhedral graph as a deltahedron, i.e., a polyhedron with congruent equilateral triangles as faces. Our experimental result shows that there are graphs that are not realizable as deltahedra. We provide an example of non-realizable graphs which are obtained by trying to construct deltahedra from each of the simple cubic polyhedral graphs with up to 10 vertices. We also show that the infinite families of non-realizable graphs can be obtained by solving the graph isomorphism problem.

Keywords: Deltahedron, polyhedral graph, geometric realization.

MSC: 51M20; 05C10, 68R10, 52B05

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