On the number of components in edge unfoldings of convex polyhedra
V. V. Astakhov, A. A. Gavrilyuk
M. V. Lomonosov Moscow State University
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.
convex polyhedron, edge unfolding, dominating set
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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
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\by V.~V.~Astakhov, A.~A.~Gavrilyuk
\paper On the number of components in edge unfoldings of convex polyhedra
\jour Model. Anal. Inform. Sist.
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