Coverage with k-transmitters in the presence of obstacles
COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part II
Coverage with k-transmitters in the presence of obstacles
Journal of Combinatorial Optimization
Unsolved problems in visibility graphs of points, segments, and polygons
ACM Computing Surveys (CSUR)
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A bar visibility representation of a graph $G$ is a collection of horizontal bars in the plane corresponding to the vertices of $G$ such that two vertices are adjacent if and only if the corresponding bars can be joined by an unobstructed vertical line segment. In a bar $k$-visibility graph, two vertices are adjacent if and only if the corresponding bars can be joined by a vertical line segment that intersects at most $k$ other bars. Bar $k$-visibility graphs were introduced by Dean et al. [J. Graph Algorithms Appl., 11 (2007), pp. 45-59]. In this paper, we present sharp upper bounds on the maximum number of edges in a bar $k$-visibility graph on $n$ vertices and the largest order of a complete bar $k$-visibility graph. We also discuss regular bar $k$-visibility graphs and forbidden induced subgraphs of bar $k$-visibility graphs.