On polygons enclosing point sets II
Let R and B be disjoint point sets such that $R\cup B$ is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P. In this paper we prove that if the vertices of the convex hull of $R\cup B$ belong...
| Autores: | , , , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2009 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/9618 |
| Acceso en línea: | https://hdl.handle.net/2117/9618 https://dx.doi.org/10.1007/s00373-009-0848-6 |
| Access Level: | acceso abierto |
| Palabra clave: | Polygons Convex geometry Graph theory Polígons Geometria convexa Grafs, Teoria de Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria convexa i discreta Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Combinatòria |
| Sumario: | Let R and B be disjoint point sets such that $R\cup B$ is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P. In this paper we prove that if the vertices of the convex hull of $R\cup B$ belong to B, and |R| ≤ |Conv(B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices $\emph{p_1}$,...,$\emph{p_n}$ and S a set of m points contained in the interior of P, m ≤ n−1. Then there is a convex decomposition {$P_1$,...,$P_n$} of P such that all points from S lie on the boundaries of $P_1$,...,$P_n$, and each $P_i$ contains a whole edge of P on its boundary. |
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