20 research outputs found
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
Segment representations with small resolution
A segment representation of a graph is an assignment of line segments in 2D
to the vertices in such a way that two segments intersect if and only if the
corresponding vertices are adjacent. Not all graphs have such segment
representations, but they exist, for example, for all planar graphs.
In this note, we study the resolution that can be achieved for segment
representations, presuming the ends of segments must be on integer grid points.
We show that any planar graph (and more generally, any graph that has a
so-called -representation) has a segment representation in a grid of width
and height
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
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