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Abstract We examine several types of visibility graphs in which sightlines can pass through k objects. For k ≥ 1 we bound the maximum thickness of semibar kvisibility graphs between ⎡2/3(k + 1) ⎤ and 2k. In addition we show that the maximum number of edges in arc and circle kvisibility graphs on n vertices is at most (k+1)(3n−k−2) for n > 4k+4 and (^{n} _{2}) for n ≤ 4k+4, while the maximum chromatic number is at most 6k+6. In semiarc kvisibility graphs on n vertices, we show that the maximum number of edges is (^{n} _{2}) for n ≤ 3k+3 and at most (k+1)(2n−(k+2)/2) for n > 3k+3, while the maximum chromatic number is at most 4k+4.

Submitted: November 2014.
Reviewed: June 2015.
Revised: June 2015.
Accepted: July 2015.
Final: July 2015.
Published: July 2015.
Communicated by
Henk Meijer
