# Obstacle Numbers of Planar Graphs

## TL;DR

This paper investigates the maximum obstacle number for planar graphs, establishing it as n-3 for an n-vertex graph, with maximal bipartite planar graphs attaining this bound, contrasting with the lower obstacle number in related classes.

## Contribution

It proves the maximum planar obstacle number of a planar graph of order n is n-3, highlighting a key difference from the standard obstacle number.

## Key findings

- Maximum planar obstacle number is n-3 for n-vertex graphs.
- Maximal bipartite planar graphs attain the maximum obstacle number.
- Obstacle number of bipartite planar graphs is 1 for graphs with at least 3 vertices.

## Abstract

Given finitely many connected polygonal obstacles $O_1,\dots,O_k$ 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 $O_1,\dots,O_k$ 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 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 $1$.