Abstract
A graph is called a bar 1-visibility graph, if its vertices can be represented as horizontal vertex-segments, called bars, and each edge corresponds to a vertical line of sight which can traverse another bar. If all bars are aligned at one side, then the graph is an aligned bar 1-visibility graph, \(AB1V\) graph for short.
We investigate \(AB1V\) graphs from different angles. First, there is a difference between maximal and optimal \(AB1V\) graphs, where optimal \(AB1V\) graphs have the maximum of \(4n-10\) edges. We show that optimal \(AB1V\) graphs can be recognized in \(\mathcal {O}(n^2)\) time and prove that an \(AB1V\) representation is fully determined by either an ordering of the bars or by the length of the bars. Moreover, we explore the relations to other classes of beyond planar graphs and show that every outer 1-planar graph is a weak \(AB1V\) graph, whereas \(AB1V\) graphs are incomparable, e.g., with planar, k-planar, outer-fan-planar, (1, j)-visibility, and RAC graphs. For the latter proofs we also use a new operation, called path-addition, which distinguishes classes of beyond planar graphs.
Supported by the Deutsche Forschungsgemeinschaft (DFG), grant Br835/18-2.
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Brandenburg, F.J., Esch, A., Neuwirth, D. (2016). On Aligned Bar 1-Visibility Graphs. In: Kaykobad, M., Petreschi, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2016. Lecture Notes in Computer Science(), vol 9627. Springer, Cham. https://doi.org/10.1007/978-3-319-30139-6_8
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DOI: https://doi.org/10.1007/978-3-319-30139-6_8
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