On aligned bar 1-visibility graphs

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 O(n2) 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 pathaddition, which distinguishes classes of beyond planar graphs.