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DOI: 10.7155/jgaa.00136
Bar kVisibility Graphs
Alice M. Dean,
William Evans,
Ellen Gethner,
Joshua D. Laison,
Mohammad Ali Safari, and
William T. Trotter
Vol. 11, no. 1, pp. 4559, 2007. Regular paper.
Abstract Let 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 also consider
the thickness of bar kvisibility graphs, obtaining an upper bound of 4
when k=1, and a bound that is quadratic in k for k > 1.

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