195 research outputs found
Grid-Obstacle Representations with Connections to Staircase Guarding
In this paper, we study grid-obstacle representations of graphs where we
assign grid-points to vertices and define obstacles such that an edge exists if
and only if an -monotone grid path connects the two endpoints without
hitting an obstacle or another vertex. It was previously argued that all planar
graphs have a grid-obstacle representation in 2D, and all graphs have a
grid-obstacle representation in 3D. In this paper, we show that such
constructions are possible with significantly smaller grid-size than previously
achieved. Then we study the variant where vertices are not blocking, and show
that then grid-obstacle representations exist for bipartite graphs. The latter
has applications in so-called staircase guarding of orthogonal polygons; using
our grid-obstacle representations, we show that staircase guarding is
\textsc{NP}-hard in 2D.Comment: To appear in the proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Minimal energy packings and collapse of sticky tangent hard-sphere polymers
We enumerate all minimal energy packings (MEPs) for small single linear and
ring polymers composed of spherical monomers with contact attractions and
hard-core repulsions, and compare them to corresponding results for monomer
packings. We define and identify ``dividing surfaces" in polymer packings,
which reduce the number of arrangements that satisfy hard-sphere and covalent
bond constraints. Compared to monomer MEPs, polymer MEPs favor intermediate
structural symmetry over high and low symmetries. We also examine the
packing-preparation dependence for longer single chains using molecular
dynamics simulations. For slow temperature quenches, chains form crystallites
with close-packed cores. As quench rate increases, the core size decreases and
the exterior becomes more disordered. By examining the contact number, we
connect suppression of crystallization to the onset of isostaticity in
disordered packings. These studies represent a significant step forward in our
ability to predict how the structural and mechanical properties of compact
polymers depend on collapse dynamics.Comment: Supplementary material is integrated in this versio
Knuthian Drawings of Series-Parallel Flowcharts
Inspired by a classic paper by Knuth, we revisit the problem of drawing
flowcharts of loop-free algorithms, that is, degree-three series-parallel
digraphs. Our drawing algorithms show that it is possible to produce Knuthian
drawings of degree-three series-parallel digraphs with good aspect ratios and
small numbers of edge bends.Comment: Full versio
Locked and Unlocked Polygonal Chains in 3D
In this paper, we study movements of simple polygonal chains in 3D. We say
that an open, simple polygonal chain can be straightened if it can be
continuously reconfigured to a straight sequence of segments in such a manner
that both the length of each link and the simplicity of the chain are
maintained throughout the movement. The analogous concept for closed chains is
convexification: reconfiguration to a planar convex polygon. Chains that cannot
be straightened or convexified are called locked. While there are open chains
in 3D that are locked, we show that if an open chain has a simple orthogonal
projection onto some plane, it can be straightened. For closed chains, we show
that there are unknotted but locked closed chains, and we provide an algorithm
for convexifying a planar simple polygon in 3D with a polynomial number of
moves.Comment: To appear in Proc. 10th ACM-SIAM Sympos. Discrete Algorithms, Jan.
199
EPG-representations with small grid-size
In an EPG-representation of a graph each vertex is represented by a path
in the rectangular grid, and is an edge in if and only if the paths
representing an share a grid-edge. Requiring paths representing edges
to be x-monotone or, even stronger, both x- and y-monotone gives rise to three
natural variants of EPG-representations, one where edges have no monotonicity
requirements and two with the aforementioned monotonicity requirements. The
focus of this paper is understanding how small a grid can be achieved for such
EPG-representations with respect to various graph parameters.
We show that there are -edge graphs that require a grid of area
in any variant of EPG-representations. Similarly there are
pathwidth- graphs that require height and area in
any variant of EPG-representations. We prove a matching upper bound of
area for all pathwidth- graphs in the strongest model, the one where edges
are required to be both x- and y-monotone. Thus in this strongest model, the
result implies, for example, , and area bounds
for bounded pathwidth graphs, bounded treewidth graphs and all classes of
graphs that exclude a fixed minor, respectively. For the model with no
restrictions on the monotonicity of the edges, stronger results can be achieved
for some graph classes, for example an area bound for bounded treewidth
graphs and bound for graphs of bounded genus.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Aligned Drawings of Planar Graphs
Let be a graph that is topologically embedded in the plane and let
be an arrangement of pseudolines intersecting the drawing of .
An aligned drawing of and is a planar polyline drawing
of with an arrangement of lines so that and are
homeomorphic to and . We show that if is
stretchable and every edge either entirely lies on a pseudoline or it has
at most one intersection with , then and have a
straight-line aligned drawing. In order to prove this result, we strengthen a
result of Da Lozzo et al., and prove that a planar graph and a single
pseudoline have an aligned drawing with a prescribed convex
drawing of the outer face. We also study the less restrictive version of the
alignment problem with respect to one line, where only a set of vertices is
given and we need to determine whether they can be collinear. We show that the
problem is NP-complete but fixed-parameter tractable.Comment: Preliminary work appeared in the Proceedings of the 25th
International Symposium on Graph Drawing and Network Visualization (GD 2017
A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem
The clustered planarity problem (c-planarity) asks whether a hierarchically
clustered graph admits a planar drawing such that the clusters can be nicely
represented by regions. We introduce the cd-tree data structure and give a new
characterization of c-planarity. It leads to efficient algorithms for
c-planarity testing in the following cases. (i) Every cluster and every
co-cluster (complement of a cluster) has at most two connected components. (ii)
Every cluster has at most five outgoing edges.
Moreover, the cd-tree reveals interesting connections between c-planarity and
planarity with constraints on the order of edges around vertices. On one hand,
this gives rise to a bunch of new open problems related to c-planarity, on the
other hand it provides a new perspective on previous results.Comment: 17 pages, 2 figure
On Upward Drawings of Trees on a Given Grid
Computing a minimum-area planar straight-line drawing of a graph is known to
be NP-hard for planar graphs, even when restricted to outerplanar graphs.
However, the complexity question is open for trees. Only a few hardness results
are known for straight-line drawings of trees under various restrictions such
as edge length or slope constraints. On the other hand, there exist
polynomial-time algorithms for computing minimum-width (resp., minimum-height)
upward drawings of trees, where the height (resp., width) is unbounded.
In this paper we take a major step in understanding the complexity of the
area minimization problem for strictly-upward drawings of trees, which is one
of the most common styles for drawing rooted trees. We prove that given a
rooted tree and a grid, it is NP-hard to decide whether
admits a strictly-upward (unordered) drawing in the given grid.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
How to compare arc-annotated sequences: The alignment hierarchy
International audienceWe describe a new unifying framework to express comparison of arc-annotated sequences, which we call alignment of arc-annotated sequences. We first prove that this framework encompasses main existing models, which allows us to deduce complexity results for several cases from the literature. We also show that this framework gives rise to new relevant problems that have not been studied yet. We provide a thorough analysis of these novel cases by proposing two polynomial time algorithms and an NP-completeness proof. This leads to an almost exhaustive study of alignment of arc-annotated sequences
Simultaneous Orthogonal Planarity
We introduce and study the problem: Given planar
graphs each with maximum degree 4 and the same vertex set, do they admit an
OrthoSEFE, that is, is there an assignment of the vertices to grid points and
of the edges to paths on the grid such that the same edges in distinct graphs
are assigned the same path and such that the assignment induces a planar
orthogonal drawing of each of the graphs?
We show that the problem is NP-complete for even if the shared
graph is a Hamiltonian cycle and has sunflower intersection and for
even if the shared graph consists of a cycle and of isolated vertices. Whereas
the problem is polynomial-time solvable for when the union graph has
maximum degree five and the shared graph is biconnected. Further, when the
shared graph is biconnected and has sunflower intersection, we show that every
positive instance has an OrthoSEFE with at most three bends per edge.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
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