Obstacle Numbers of Planar Graphs [article]

John Gimbel, Patrice Ossona de Mendez, Pavel Valtr
2017 arXiv   pre-print
Given finitely many connected polygonal obstacles O_1,...,O_k in the plane and a set P of points in general position and not in any obstacle, the visibility graph of P with obstacles O_1,...,O_k 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
more » ... lanar 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.
arXiv:1706.06992v3 fatcat:ucgmzezbazd6rhqwfborctcjtq