A geometric graph G is a graph whose vertex set is a set Pn of n points on the plane in general position, and whose edges are straight line segments (which may cross) joining pairs of vertices of G. We say that G contains a convex r-gon if its vertex and edge sets contain, respectively, the vertices and edges of a convex polygon with r vertices. In this paper we study the following problem: Which is the largest number of edges that a geometric graph with n vertices may have in such a way that it does not contain a convex r-gon? We give sharp bounds for this problem. We also give some bounds for the following problem: Given a point set, how many edges can a geometric graph with vertex set Pn have such that it does not contain a convex r-gon? A result of independent interest is also proved here, namely: Let Pn be a set of n points in general position. Then there are always three concurrent lines such that each of the six wedges defined by the lines contains exactly [n/6] or [n/6] elements of Pn. © Springer-Verlag Berlin Heidelberg 2003.
CITATION STYLE
Nara, C., Sakai, T., & Urrutia, J. (2003). Maximal number of edges in geometric graphs without convex polygons. Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2866, 215–220. https://doi.org/10.1007/978-3-540-44400-8_23
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