19 research outputs found
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
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
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
Distinguishing Classes of Intersection Graphs of Homothets or Similarities of Two Convex Disks
For smooth convex disks A, i.e., convex compact subsets of the plane with non-empty interior, we classify the classes G^{hom}(A) and G^{sim}(A) of intersection graphs that can be obtained from homothets and similarities of A, respectively. Namely, we prove that G^{hom}(A) = G^{hom}(B) if and only if A and B are affine equivalent, and G^{sim}(A) = G^{sim}(B) if and only if A and B are similar