Given finitely many connected polygonal obstacles O1,…,Ok in the
plane and a set P of points in general position and not in any obstacle, the
{\em visibility graph} of P with obstacles O1,…,Ok is the (geometric)
graph with vertex set P, where two vertices are adjacent if the straight line
segment joining them intersects no obstacle. The obstacle number of a graph G
is the smallest integer k such that G is the visibility graph of a set of
points with k obstacles. If G is planar, we define the planar obstacle
number of G by further requiring that the visibility graph has no crossing
edges (hence that it is a planar geometric drawing of G). In this paper, we
prove that the maximum planar obstacle number of a planar graph of order n is
n−3, the maximum being attained (in particular) by maximal bipartite planar
graphs. This displays a significant difference with the standard obstacle
number, as we prove that the obstacle number of every bipartite planar graph
(and more generally in the class PURE-2-DIR of intersection graphs of straight
line segments in two directions) of order at least 3 is 1.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017