Bar kVisibility Graphs: Bounds on the Number of Edges, Chromatic Number, and ThicknessDean, Alice M. and Evans, William S. and Gethner, Ellen and Laison, Joshua D. and Safari, Mohammad Ali and Trotter, William T. (2006) Bar kVisibility Graphs: Bounds on the Number of Edges, Chromatic Number, and Thickness. In: Graph Drawing 13th International Symposium, GD 2005, September 1214, 2005 , pp. 7382(Official URL: http://dx.doi.org/10.1007/11618058_7). Full text not available from this repository.
Official URL: http://dx.doi.org/10.1007/11618058_7
AbstractLet S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a onetoone correspondence between vertices of G and bars in S, such that there is an edge between two vertices in G if and only if there exists an unobstructed vertical line of sight between their corresponding bars. If bars are allowed to see through each other, the graphs representable in this way are precisely the interval graphs. We consider representations in which bars are allowed to see through at most k other bars. Since all bar visibility graphs are planar, we seek measurements of closeness to planarity for bar kvisibility graphs. We obtain an upper bound on the number of edges in a bar kvisibility graph. As a consequence, we obtain an upper bound of 12 on the chromatic number of bar 1visibility graphs, and a tight upper bound of 8 on the size of the largest complete bar 1visibility graph. We conjecture that bar 1visibility graphs have thickness at most 2.
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