5 research outputs found
Contact graphs of line segments are NP-complete
AbstractContact graphs are a special kind of intersection graphs of geometrical objects in which the objects are not allowed to cross but only to touch each other. Contact graphs of line segments in the plane are considered — it is proved that recognizing line-segment contact graphs, with contact degrees of 3 or more, is an NP-complete problem, even for planar graphs. This result contributes to the related research on recognition complexity of curve contact graphs (Hliněný J. Combin. Theory Ser. B 74 (1998) 87)
Coloring non-crossing strings
For a family of geometric objects in the plane
, define as the least
integer such that the elements of can be colored with
colors, in such a way that any two intersecting objects have distinct
colors. When is a set of pseudo-disks that may only intersect on
their boundaries, and such that any point of the plane is contained in at most
pseudo-disks, it can be proven that
since the problem is equivalent to cyclic coloring of plane graphs. In this
paper, we study the same problem when pseudo-disks are replaced by a family
of pseudo-segments (a.k.a. strings) that do not cross. In other
words, any two strings of are only allowed to "touch" each other.
Such a family is said to be -touching if no point of the plane is contained
in more than elements of . We give bounds on
as a function of , and in particular we show that
-touching segments can be colored with colors. This partially answers
a question of Hlin\v{e}n\'y (1998) on the chromatic number of contact systems
of strings.Comment: 19 pages. A preliminary version of this work appeared in the
proceedings of EuroComb'09 under the title "Coloring a set of touching
strings
Characterising circular-arc contact -VPG graphs
A contact -VPG graph is a graph for which there exists a collection of
nontrivial pairwise interiorly disjoint horizontal and vertical segments in
one-to-one correspondence with its vertex set such that two vertices are
adjacent if and only if the corresponding segments touch. It was shown by Deniz
et al. that Recognition is -complete for contact -VPG graphs.
In this paper we present a minimal forbidden induced subgraph characterisation
of contact -VPG graphs within the class of circular-arc graphs and provide
a polynomial-time algorithm for recognising these graphs
On contact graphs of paths on a grid
In this paper we consider Contact graphs of Paths on a Grid (CPG graphs), i.e. graphs for which there exists a family of interiorly disjoint paths on a grid in one-to-one correspondence with their vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. Our class generalizes the well studied class of VCPG graphs (see [1]). We examine CPG graphs from a structural point of view which leads to constant upper bounds on the clique number and the chromatic number. Moreover, we investigate the recognition and 3-colorability problems for B0- CPG, a subclass of CPG. We further show that CPG graphs are not necessarily planar and not all planar graphs are CPG