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.
['Stephen G. Hartke', 'Jennifer Vandenbussche', 'Paul S. Wenger']
Further Results on Bar k-Visibility Graphs