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HOW TO CUT OUT A CONVEX POLYHEDRON
No full textIt is known that one can fold a convex polyhedron from a non-overlapping face unfolding, but the complexity of the algorithm in [MP] remains an open prob-lem. In this paper we show that every convex polyhedron P ⊂ Rd can be obtained in polynomial time, by starting with a cube which contains P and sequentially cutting out the extra parts of the surface. Our main tool is of independent interest. We prove that given a convex polytope P in Rd and a facet F of P, F is contained in the union ∪G 6=FΦF,G(G). The union is over all the facets G of P different from F and ΦF,G(G) is the set obtained from G by rotating the hyperplane which contains G about the intersection of it with the hyperplane which contains F until they coincide
Regular matchstick graphs
Get PDFAbstract A match-stick graph is a plane geometric graph in which every edge has length 1 and no two edges cross each other. It was conjectured that no 5-regular match-stick graph exists. In this paper we prove this conjecture
NOTE ON THE NUMBER OF EDGES IN FAMILIES WITH LINEAR UNION-COMPLEXITY
No full textWe give a simple argument showing that the number of edges in the intersection graph G of a family of n sets in the plane with a linear union-complexity is O(ω(G)n). In particular, we prove χ(G) 6 col(G) < 19ω(G) for intersection graph G of a family of pseudo-discs, which improves a previous bound
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