On bar (1, j)-visibility graphs

Abstract

A graph is called a bar (1, j)-visibility graph if its vertices can be represented as horizontal vertex-segments (bars) and each edge as a vertical edge-segment connecting the bars of the end vertices such that each edge-segment intersects at most one other bar and each bar is intersected by at most j edge-segments. Bar (1, j)-visibility refines bar 1-visibility in which there is no bound on the number of intersections of bars. We construct gadgets which show structural properties of bar (1, j)- visibility graphs, study bounds on the maximal number of edges and show that there is an infinite hierarchy of bar (1, j)-visibility graphs. Finally, we prove that it is NP-complete to test whether a graph is bar (1,∞)-visible.