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 Model. Anal. Inform. Sist., 2009, Volume 16, Number 1, Pages 16–23 (Mi mais45)

On the number of components in edge unfoldings of convex polyhedra

V. V. Astakhov, A. A. Gavrilyuk

M. V. Lomonosov Moscow State University

Abstract: In the theory of convex polyhedra there is a problem left unsolved which is sometimes called The Durer problem: Does every convex polyhedron have at least one connected unfolding? In this paper we consider a related problem: Find the upper bound $c(P)$ for the number of components in the edge unfolding of a convex polyhedron $P$ in terms of the number $F$ of faces. We showed that $c(P)$ does not exceed the cardinality of any dominating set in the dual graph $G(P)$ of the polyhedron $P$. Next we proved that there exists a dominating set in $G(P)$ of cardinality not more than $3F/7$. These two statements lead to an estimation $c(P)\le 3F/7$ that was proved in this work.

Keywords: convex polyhedron, edge unfolding, dominating set

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UDC: 519.17

Citation: V. V. Astakhov, A. A. Gavrilyuk, “On the number of components in edge unfoldings of convex polyhedra”, Model. Anal. Inform. Sist., 16:1 (2009), 16–23

Citation in format AMSBIB
\Bibitem{AstGav09} \by V.~V.~Astakhov, A.~A.~Gavrilyuk \paper On the number of components in edge unfoldings of convex polyhedra \jour Model. Anal. Inform. Sist. \yr 2009 \vol 16 \issue 1 \pages 16--23 \mathnet{http://mi.mathnet.ru/mais45}