Thickness of Bar 1-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.  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. For a special case of bar 1-visibility graphs we present an algorithm partitioning the edges into two plane graphs, showing that for this class the thickness is indeed bounded by 2.
KeywordsPlanar Graph Chromatic Number Interval Graph Horizontal Part Graph Draw
- 3.Dean, A.M., Evans, W., Gethner, E., Laison, J.D., Safari, M.A., Trotter, W.T.: Bar k-visibility graphs. Manuscript (2005)Google Scholar
- 5.Dean, A.M., Gethner, E., Hutchinson, J.P.: Unit bar-visibility layouts of triangulated polygons. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 111–121. Springer, Heidelberg (2005)Google Scholar
- 8.Hartke, S.G., Vandenbussche, J., Wenger, P.: Further results on bar k-visibility graphs. Manuscript (November 2005)Google Scholar