Motivated by questions in computer vision and sensor networks, Alpert et al. [3] introduced the following definitions. Given a graph G, an obstacle representation of G is a set of points in the plane representing the vertices of G, together with a set of connected obstacles such that two vertices of G are joined by an edge if an only if the corresponding points can be connected by a segment which avoids all obstacles. The obstacle number of G is the minimum number of obstacles in an obstacle representation of G. It was shown in [3] that there exist graphs of n vertices with obstacle number at least Ω(√logn). We use extremal graph theoretic tools to show that (1) there exist graphs of n vertices with obstacle number at least Ω(n/log2 n), and (2) the total number of graphs on n vertices with bounded obstacle number is at most 20(n2). Better results are proved if we are allowed to use only convex obstacles or polygonal obstacles with a small number of sides. © 2010 Springer-Verlag.
CITATION STYLE
Mukkamala, P., Pach, J., & Sariöz, D. (2010). Graphs with large obstacle numbers. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6410 LNCS, pp. 292–303). https://doi.org/10.1007/978-3-642-16926-7_27
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