Parameters of Bar k-Visibility Graphs

Bar k-visibility graphs are graphs admitting a representation in which the vertices correspond to horizontal line segments, called bars, and the edges correspond to vertical lines of sight which can traverse up to k bars. These graphs were introduced by Dean et al. [4] who conjectured that bar 1-visibility graphs have thickness at most 2. We construct a bar 1-visibility graph having thickness 3, disproving their conjecture. Furthermore, we define semi bar k-visibility graphs, a subclass of bar k-visibility graphs, and show tight results for a number of graph parameters including chromatic number, maximum number of edges and connectivity. Then we present an algorithm partitioning the edges of a semi bar 1-visibility graph into two plane graphs, showing that for this subclass the (geometric) thickness is indeed bounded by 2. Article Type Communicated by Submitted Revised Regular paper M. Kaufmann and D. Wagner January 2007 August 2007 Research of M. Massow supported by the Research Training Group “Methods for Discrete Structures” (DFG-GRK 1408) and the Berlin Mathematical School. S. Felsner and M. Massow, Parameters of BkVs, JGAA, 12(1) 5–27 (2008) 6