On k-visibility graphs - Babbitt, Matthew; Geneson, Jesse; Khovanova, Tanya (Journal of Graph Algorithms and Applications, 2015)
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 ⎡2/3(k + 1) ⎤ 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.