Discrete Applied Mathematics - Combinatorics and complexity
Excluding induced subgraphs II: extremal graphs
Discrete Applied Mathematics
Handbook of discrete and computational geometry
Visibility Algorithms in the Plane
Visibility Algorithms in the Plane
Blocking Visibility for Points in General Position
Discrete & Computational Geometry - Special Issue Dedicated to the Memory of Victor Klee
Unsolved problems in visibility graphs of points, segments, and polygons
ACM Computing Surveys (CSUR)
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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 2o(n2). Better results are proved if we are allowed to use only convex obstacles or polygonal obstacles with a small number of sides.