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 semi-bar k-visibility graphs between d 2 (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 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.
Highlights
Visibility graphs are graphs for which vertices can be drawn as regions so that two regions are visible to each other if and only if there is an edge between their corresponding vertices
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 and at most
Summary
Visibility graphs are graphs for which vertices can be drawn as regions so that two regions are visible to each other if and only if there is an edge between their corresponding vertices. In this paper we study bar, semi-bar, arc, circle, and semi-arc visibility graphs. We study a variant of visibility graphs represented by drawings in which objects are able to see through exactly k other objects for some positive integer k. Dean et al [1] previously placed upper bounds on the number of edges, the chromatic number, and the thickness of bar k-visibility graphs with n vertices in terms of k and n. Hartke et al [4] found sharp upper bounds on the maximum number of edges in bar k-visibility graphs. We bound the maximum number of edges and the chromatic numbers of arc, circle, and semi-arc k-visibility graphs. The proofs in this paper do not use the results claimed in [11]
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