65 research outputs found
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
Contact Representations of Graphs in 3D
We study contact representations of graphs in which vertices are represented
by axis-aligned polyhedra in 3D and edges are realized by non-zero area common
boundaries between corresponding polyhedra. We show that for every 3-connected
planar graph, there exists a simultaneous representation of the graph and its
dual with 3D boxes. We give a linear-time algorithm for constructing such a
representation. This result extends the existing primal-dual contact
representations of planar graphs in 2D using circles and triangles. While
contact graphs in 2D directly correspond to planar graphs, we next study
representations of non-planar graphs in 3D. In particular we consider
representations of optimal 1-planar graphs. A graph is 1-planar if there exists
a drawing in the plane where each edge is crossed at most once, and an optimal
n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a
linear-time algorithm for representing optimal 1-planar graphs without
separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph
admits a representation with boxes. Hence, we consider contact representations
with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a
quadratic-time algorithm for representing optimal 1-planar graph with L-shaped
polyhedra
Morphing Contact Representations of Graphs
We consider the problem of morphing between contact representations of a plane graph. In a contact representation of a plane graph, vertices are realized by internally disjoint elements from a family of connected geometric objects. Two such elements touch if and only if their corresponding vertices are adjacent. These touchings also induce the same embedding as in the graph. In a morph between two contact representations we insist that at each time step (continuously throughout the morph) we have a contact representation of the same type.
We focus on the case when the geometric objects are triangles that are the lower-right half of axis-parallel rectangles. Such RT-representations exist for every plane graph and right triangles are one of the simplest families of shapes supporting this property. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs.
We study piecewise linear morphs, where each step is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. We provide a polynomial-time algorithm that decides whether there is a piecewise linear morph between two RT-representations of a plane triangulation, and, if so, computes a morph with a quadratic number of linear morphs. As a direct consequence, we obtain that for 4-connected plane triangulations there is a morph between every pair of RT-representations where the "top-most" triangle in both representations corresponds to the same vertex. This shows that the realization space of such RT-representations of any 4-connected plane triangulation forms a connected set
Adjacency Graphs of Polyhedral Surfaces
We study whether a given graph can be realized as an adjacency graph of the
polygonal cells of a polyhedral surface in . We show that every
graph is realizable as a polyhedral surface with arbitrary polygonal cells, and
that this is not true if we require the cells to be convex. In particular, if
the given graph contains , , or any nonplanar -tree as a
subgraph, no such realization exists. On the other hand, all planar graphs,
, and can be realized with convex cells. The same holds for
any subdivision of any graph where each edge is subdivided at least once, and,
by a result from McMullen et al. (1983), for any hypercube.
Our results have implications on the maximum density of graphs describing
polyhedral surfaces with convex cells: The realizability of hypercubes shows
that the maximum number of edges over all realizable -vertex graphs is in
. From the non-realizability of , we obtain that
any realizable -vertex graph has edges. As such, these graphs
can be considerably denser than planar graphs, but not arbitrarily dense.Comment: To appear in Proc. SoCG 202
Cover Contact Graphs.
We study problems that arise in the context of covering certain geometric objects (so-called seeds, e.g., points or disks) by a set of other geometric objects (a so-called cover, e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pair wise disjoint, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in two types of tasks: (a) deciding whether a given seed set has a connected CCG, and (b) deciding whether a given graph has a realization as a CCG on a given seed set. Concerning task (a) we give efficient algorithms for the case that seeds are points and covers are disks or triangles. We show that the problem becomes NP-hard if seeds and covers are disks. Concerning task (b) we show that it is even NP-hard for point seeds and disk covers (given a fixed correspondence between vertices and seeds)
Cover Contact Graphs
Es una ponencia presentada al 15th International Symposium on Graph Drawing (2007)We study problems that arise in the context of covering certain geometric objects (so-called seeds, e.g., points or disks) by a set of other geometric objects (a so-called cover, e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in two types of tasks: (a) deciding whether a given seed set has a connected CCG, and (b) deciding whether a given graph has a realization as a CCG on a given seed set. Concerning task (a) we give efficient algorithms for the case that seeds are points and covers are disks or triangles. We show that the problem becomes NP-hard if seeds and covers are disks. Concerning task (b) we show that it is even NP-hard for point seeds and disk covers (given a fixed correspondence between vertices and seeds).German Research Foundation WO 758/4-
Periodic Auxetics: Structure and Design
Materials science has adopted the term of auxetic behavior for structural deformations where stretching in some direction entails lateral widening, rather than lateral shrinking. Most studies, in the last three decades, have explored repetitive or cellular structures and used the notion of negative Poisson\u27s ratio as the hallmark of auxetic behavior. However, no general auxetic principle has been established from this perspective. In the present article, we show that a purely geometric approach to periodic auxetics is apt to identify essential characteristics of frameworks with auxetic deformations and can generate a systematic and endless series of periodic auxetic designs. The critical features refer to convexity properties expressed through families of homothetic ellipsoids
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