3,610 research outputs found
On Visibility Representations of Non-planar Graphs
A rectangle visibility representation (RVR) of a graph consists of an
assignment of axis-aligned rectangles to vertices such that for every edge
there exists a horizontal or vertical line of sight between the rectangles
assigned to its endpoints. Testing whether a graph has an RVR is known to be
NP-hard. In this paper, we study the problem of finding an RVR under the
assumption that an embedding in the plane of the input graph is fixed and we
are looking for an RVR that reflects this embedding. We show that in this case
the problem can be solved in polynomial time for general embedded graphs and in
linear time for 1-plane graphs (i.e., embedded graphs having at most one
crossing per edge). The linear time algorithm uses a precise list of forbidden
configurations, which extends the set known for straight-line drawings of
1-plane graphs. These forbidden configurations can be tested for in linear
time, and so in linear time we can test whether a 1-plane graph has an RVR and
either compute such a representation or report a negative witness. Finally, we
discuss some extensions of our study to the case when the embedding is not
fixed but the RVR can have at most one crossing per edge
Witness (Delaunay) Graphs
Proximity graphs are used in several areas in which a neighborliness
relationship for input data sets is a useful tool in their analysis, and have
also received substantial attention from the graph drawing community, as they
are a natural way of implicitly representing graphs. However, as a tool for
graph representation, proximity graphs have some limitations that may be
overcome with suitable generalizations. We introduce a generalization, witness
graphs, that encompasses both the goal of more power and flexibility for graph
drawing issues and a wider spectrum for neighborhood analysis. We study in
detail two concrete examples, both related to Delaunay graphs, and consider as
well some problems on stabbing geometric objects and point set discrimination,
that can be naturally described in terms of witness graphs.Comment: 27 pages. JCCGG 200
Witness Gabriel Graphs
We consider a generalization of the Gabriel graph, the witness Gabriel graph.
Given a set of vertices P and a set of witnesses W in the plane, there is an
edge ab between two points of P in the witness Gabriel graph GG-(P,W) if and
only if the closed disk with diameter ab does not contain any witness point
(besides possibly a and/or b). We study several properties of the witness
Gabriel graph, both as a proximity graph and as a new tool in graph drawing.Comment: 23 pages. EuroCG 200
Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible
We analyze the computational complexity of the many types of
pencil-and-paper-style puzzles featured in the 2016 puzzle video game The
Witness. In all puzzles, the goal is to draw a simple path in a rectangular
grid graph from a start vertex to a destination vertex. The different puzzle
types place different constraints on the path: preventing some edges from being
visited (broken edges); forcing some edges or vertices to be visited
(hexagons); forcing some cells to have certain numbers of incident path edges
(triangles); or forcing the regions formed by the path to be partially
monochromatic (squares), have exactly two special cells (stars), or be singly
covered by given shapes (polyominoes) and/or negatively counting shapes
(antipolyominoes). We show that any one of these clue types (except the first)
is enough to make path finding NP-complete ("witnesses exist but are hard to
find"), even for rectangular boards. Furthermore, we show that a final clue
type (antibody), which necessarily "cancels" the effect of another clue in the
same region, makes path finding -complete ("witnesses do not exist"),
even with a single antibody (combined with many anti/polyominoes), and the
problem gets no harder with many antibodies. On the positive side, we give a
polynomial-time algorithm for monomino clues, by reducing to hexagon clues on
the boundary of the puzzle, even in the presence of broken edges, and solving
"subset Hamiltonian path" for terminals on the boundary of an embedded planar
graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of
this paper appeared at the 9th International Conference on Fun with
Algorithms (FUN 2018
New Classes of Distributed Time Complexity
A number of recent papers -- e.g. Brandt et al. (STOC 2016), Chang et al.
(FOCS 2016), Ghaffari & Su (SODA 2017), Brandt et al. (PODC 2017), and Chang &
Pettie (FOCS 2017) -- have advanced our understanding of one of the most
fundamental questions in theory of distributed computing: what are the possible
time complexity classes of LCL problems in the LOCAL model? In essence, we have
a graph problem in which a solution can be verified by checking all
radius- neighbourhoods, and the question is what is the smallest such
that a solution can be computed so that each node chooses its own output based
on its radius- neighbourhood. Here is the distributed time complexity of
.
The time complexity classes for deterministic algorithms in bounded-degree
graphs that are known to exist by prior work are , , , , and . It is also known
that there are two gaps: one between and , and
another between and . It has been conjectured
that many more gaps exist, and that the overall time hierarchy is relatively
simple -- indeed, this is known to be the case in restricted graph families
such as cycles and grids.
We show that the picture is much more diverse than previously expected. We
present a general technique for engineering LCL problems with numerous
different deterministic time complexities, including
for any , for any , and
for any in the high end of the complexity
spectrum, and for any ,
for any , and
for any in the low end; here
is a positive rational number
Triangle-free intersection graphs of line segments with large chromatic number
In the 1970s, Erdos asked whether the chromatic number of intersection graphs
of line segments in the plane is bounded by a function of their clique number.
We show the answer is no. Specifically, for each positive integer , we
construct a triangle-free family of line segments in the plane with chromatic
number greater than . Our construction disproves a conjecture of Scott that
graphs excluding induced subdivisions of any fixed graph have chromatic number
bounded by a function of their clique number.Comment: Small corrections, bibliography updat
Piercing axis-parallel boxes
Let \F be a finite family of axis-parallel boxes in such that \F
contains no pairwise disjoint boxes. We prove that if \F contains a
subfamily \M of pairwise disjoint boxes with the property that for every
F\in \F and M\in \M with , either contains a
corner of or contains corners of , then \F can be
pierced by points. One consequence of this result is that if and
the ratio between any of the side lengths of any box is bounded by a constant,
then \F can be pierced by points. We further show that if for each two
intersecting boxes in \F a corner of one is contained in the other, then \F
can be pierced by at most points, and in the special case
where \F contains only cubes this bound improves to
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