Geodesic
On the number of components in edge unfoldings of convex polyhedra
Modelirovanie i analiz informacionnyh sistem, Tome 16 (2009) no. 1, pp. 16-23.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{MAIS_2009_16_1_a1,
     author = {V. V. Astakhov and A. A. Gavrilyuk},
     title = {On the number of components in edge unfoldings of convex polyhedra},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {16--23},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2009_16_1_a1/}
}
TY  - JOUR
AU  - V. V. Astakhov
AU  - A. A. Gavrilyuk
TI  - On the number of components in edge unfoldings of convex polyhedra
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2009
SP  - 16
EP  - 23
VL  - 16
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MAIS_2009_16_1_a1/
LA  - ru
ID  - MAIS_2009_16_1_a1
ER  - 
%0 Journal Article
%A V. V. Astakhov
%A A. A. Gavrilyuk
%T On the number of components in edge unfoldings of convex polyhedra
%J Modelirovanie i analiz informacionnyh sistem
%D 2009
%P 16-23
%V 16
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MAIS_2009_16_1_a1/
%G ru
%F MAIS_2009_16_1_a1
V. V. Astakhov; A. A. Gavrilyuk. On the number of components in edge unfoldings of convex polyhedra. Modelirovanie i analiz informacionnyh sistem, Tome 16 (2009) no. 1, pp. 16-23. http://geodesic.mathdoc.fr/item/MAIS_2009_16_1_a1/

[1] A. S. Tarasov, “Mnogogranniki, ne dopuskayuschie naturalnykh razvërtok”, Uspekhi matematicheskikh nauk, 54:3 (1999), 185–186 | MR | Zbl

[2] A. A. Glazyrin, A. S. Tarasov, “Anti-Dyurer — gipoteza dlya nevypuklykh mnogogrannikov”, Materialy IX Mezhdunarodnogo seminara «Diskretnaya matematika i eë prilozheniya», M., 2007

[3] Erik D. Demaine and Joseph O'Rourke, Geometric Folding Algorithms, Cambridge University Press, 2007 | MR

[4] Günter M. Ziegler, Lectures on Polytopes, Springer-Verlag, New York, 1995 | MR