Outerstring graphs are χ\chi-bounded

An outerstring graph is an intersection graph of curves that lie in a common half-plane and have one endpoint on the boundary of that half-plane. We prove that the class of outerstring graphs is $\chi$-bounded, which means that their chromatic number is bounded by a function of their clique number. This generalizes a series of previous results on $\chi$-boundedness of outerstring graphs with various additional restrictions on the shape of curves or the number of times the pairs of curves can cross. The assumption that each curve has an endpoint on the boundary of the half-plane is justified by the known fact that triangle-free intersection graphs of straight-line segments can have arbitrarily large chromatic number.Comment: Introduction extended by a survey of results on (outer)string graphs, some minor correction

Colouring Polygon Visibility Graphs and Their Generalizations

Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ? has chromatic number at most 3?4^{?-1}. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time

On grounded L-graphs and their relatives

We consider the graph class Grounded-L corresponding to graphs that admit an intersection representation by L-shaped curves, where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove that Grounded-L graphs admit an equivalent characterisation in terms of vertex ordering with forbidden patterns. We also compare this class to related intersection classes, such as the grounded segment graphs, the monotone L-graphs (a.k.a. max point-tolerance graphs), or the outer-1-string graphs. We give constructions showing that these classes are all distinct and satisfy only trivial or previously known inclusions.Comment: 16 pages, 6 figure

Coloring intersection graphs of arc-connected sets in the plane

A family of sets in the plane is simple if the intersection of its any subfamily is arc-connected, and it is pierced by a line $L$ if the intersection of its any member with $L$ is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their clique number.Comment: Minor changes + some additional references not included in the journal versio

Intersection Graphs of Rays and Grounded Segments

We consider several classes of intersection graphs of line segments in the plane and prove new equality and separation results between those classes. In particular, we show that: (1) intersection graphs of grounded segments and intersection graphs of downward rays form the same graph class, (2) not every intersection graph of rays is an intersection graph of downward rays, and (3) not every intersection graph of rays is an outer segment graph. The first result answers an open problem posed by Cabello and Jej\v{c}i\v{c}. The third result confirms a conjecture by Cabello. We thereby completely elucidate the remaining open questions on the containment relations between these classes of segment graphs. We further characterize the complexity of the recognition problems for the classes of outer segment, grounded segment, and ray intersection graphs. We prove that these recognition problems are complete for the existential theory of the reals. This holds even if a 1-string realization is given as additional input.Comment: 16 pages 12 Figure

Hasse diagrams with large chromatic number

For every positive integer $n$, we construct a Hasse diagram with $n$ vertices and chromatic number $\Omega(n^{1/4})$, which significantly improves on the previously known best constructions of Hasse diagrams having chromatic number $\Theta(\log n)$. In addition, if we also require that our Hasse diagram has girth at least $k\geq 5$, we can achieve a chromatic number of at least $n^{\frac{1}{2k-3}+o(1)}$. These results have the following surprising geometric consequence. They imply the existence of a family $\mathcal{C}$ of $n$ curves in the plane such that the disjointness graph $G$ of $\mathcal{C}$ is triangle-free (or have high girth), but the chromatic number of $G$ is polynomial in $n$. Again, the previously known best construction, due to Pach, Tardos and T\'oth, had only logarithmic chromatic number.Comment: 11 pages, 1 figur

On grounded L-graphs and their relatives

We consider the graph class Grounded-L corresponding to graphs that admit an intersection representation by L-shaped curves, where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove that Grounded-L graphs admit an equivalent characterisation in terms of vertex ordering with forbidden patterns. We also compare this class to related intersection classes, such as the grounded segment graphs, the monotone L-graphs (a.k.a. max point-tolerance graphs), or the outer-1-string graphs. We give constructions showing that these classes are all distinct and satisfy only trivial or previously known inclusions