635 research outputs found
Recognizing Geometric Intersection Graphs Stabbed by a Line
In this paper, we determine the computational complexity of recognizing two
graph classes, \emph{grounded L}-graphs and \emph{stabbable grid intersection}
graphs. An L-shape is made by joining the bottom end-point of a vertical
() segment to the left end-point of a horizontal () segment. The top
end-point of the vertical segment is known as the {\em anchor} of the L-shape.
Grounded L-graphs are the intersection graphs of L-shapes such that all the
L-shapes' anchors lie on the same horizontal line. We show that recognizing
grounded L-graphs is NP-complete. This answers an open question asked by
Jel{\'\i}nek \& T{\"o}pfer (Electron. J. Comb., 2019).
Grid intersection graphs are the intersection graphs of axis-parallel line
segments in which two vertical (similarly, two horizontal) segments cannot
intersect. We say that a (not necessarily axis-parallel) straight line
stabs a segment , if intersects . A graph is a stabbable grid
intersection graph () if there is a grid intersection representation
of in which the same line stabs all its segments. We show that recognizing
graphs is -complete, even on a restricted class of graphs. This
answers an open question asked by Chaplick \etal (\textsc{O}rder, 2018).Comment: 18 pages, 11 Figure
Pixel and Voxel Representations of Graphs
We study contact representations for graphs, which we call pixel
representations in 2D and voxel representations in 3D. Our representations are
based on the unit square grid whose cells we call pixels in 2D and voxels in
3D. Two pixels are adjacent if they share an edge, two voxels if they share a
face. We call a connected set of pixels or voxels a blob. Given a graph, we
represent its vertices by disjoint blobs such that two blobs contain adjacent
pixels or voxels if and only if the corresponding vertices are adjacent. We are
interested in the size of a representation, which is the number of pixels or
voxels it consists of.
We first show that finding minimum-size representations is NP-complete. Then,
we bound representation sizes needed for certain graph classes. In 2D, we show
that, for -outerplanar graphs with vertices, pixels are
always sufficient and sometimes necessary. In particular, outerplanar graphs
can be represented with a linear number of pixels, whereas general planar
graphs sometimes need a quadratic number. In 3D, voxels are
always sufficient and sometimes necessary for any -vertex graph. We improve
this bound to for graphs of treewidth and to
for graphs of genus . In particular, planar graphs
admit representations with voxels
- …