Planar graphs as L-intersection or L-contact graphs

The L-intersection graphs are the graphs that have a representation as intersection graphs of axis parallel shapes in the plane. A subfamily of these graphs are {L, |, --}-contact graphs which are the contact graphs of axis parallel L, |, and -- shapes in the plane. We prove here two results that were conjectured by Chaplick and Ueckerdt in 2013. We show that planar graphs are L-intersection graphs, and that triangle-free planar graphs are {L, |, --}-contact graphs. These results are obtained by a new and simple decomposition technique for 4-connected triangulations. Our results also provide a much simpler proof of the known fact that planar graphs are segment intersection graphs

1-String CZ-Representation of Planar Graphs

In this paper, we prove that every planar 4-connected graph has a CZ-representation---a string representation using paths in a rectangular grid that contain at most one vertical segment. Furthermore, two paths representing vertices $u,v$ intersect precisely once whenever there is an edge between $u$ and $v$. The required size of the grid is $n \times 2n$

Drawings of Planar Graphs with Few Slopes and Segments

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on $n$ vertices has a plane drawing with at most ${5/2}n$ segments and at most $2n$ slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.Comment: This paper is submitted to a journal. A preliminary version appeared as "Really Straight Graph Drawings" in the Graph Drawing 2004 conference. See http://arxiv.org/math/0606446 for a companion pape

Restricted frame graphs and a conjecture of Scott

Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is replaced by any graph $H$. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erd\H{o}s). This shows that Scott's conjecture is false whenever $H$ is obtained from a non-planar graph by subdividing every edge at least once. It remains interesting to decide which graphs $H$ satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from $K_4$ by subdividing every edge at least once. We also prove that if $G$ is a 2-connected multigraph with no vertex contained in every cycle of $G$, then any graph obtained from $G$ by subdividing every edge at least twice is a counterexample to Scott's conjecture.Comment: 21 pages, 8 figures - Revised version (note that we moved some of our results to an appendix

Obstacle Numbers of Planar Graphs

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 $1$.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

The Complexity of Drawing Graphs on Few Lines and Few Planes

It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the minimum number of lines in $\mathbb{R}^d$ that together can cover all edges of a drawing of $G$. For $d=2$, $G$ must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results. - For $d\in\{2,3\}$, we prove that deciding whether $\rho^1_d(G)\le k$ for a given graph $G$ and integer $k$ is ${\exists\mathbb{R}}$-complete. - Since $\mathrm{NP}\subseteq{\exists\mathbb{R}}$, deciding $\rho^1_d(G)\le k$ is NP-hard for $d\in\{2,3\}$. On the positive side, we show that the problem is fixed-parameter tractable with respect to $k$. - Since ${\exists\mathbb{R}}\subseteq\mathrm{PSPACE}$, both $\rho^1_2(G)$ and $\rho^1_3(G)$ are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to $\rho^1_2$ or $\rho^1_3$ sometimes require irrational coordinates. - Let $\rho^2_3(G)$ be the minimum number of planes in $\mathbb{R}^3$ needed to cover a straight-line drawing of a graph $G$. We prove that deciding whether $\rho^2_3(G)\le k$ is NP-hard for any fixed $k \ge 2$. Hence, the problem is not fixed-parameter tractable with respect to $k$ unless $\mathrm{P}=\mathrm{NP}$