On k-visibility graphs

We examine several types of visibility graphs in which sightlines can pass through k objects. For k ≥ 1 we bound the maximum thickness of semi-bar k-visibility graphs between d 2 3 (k + 1)e and 2k. In addition we show that the maximum number of edges in arc and circle k-visibility 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 semi-arc k-visibility 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 Article type: Regular Paper Communicated by: H. Meijer J. Geneson was supported by the NSF Graduate Research Fellowship under Grant No. 1122374. E-mail addresses: [email protected] (Matthew Babbitt) [email protected] (Jesse Geneson) [email protected] (Tanya Khovanova) 346 M. Babbitt, J. Geneson, T. Khovanova On k-visibility graphs