11 research outputs found
On the Structure of Graphs with Low Obstacle Number
The obstacle number of a graph G is the smallest number of polygonal obstacles in the plane with the property that the vertices of G can be represented by distinct points such that two of them see each other if and only if the corresponding vertices are joined by an edge. We list three small graphs that require more than one obstacle. Using extremal graph theoretic tools developed by Prömel, Steger, Bollobás, Thomason, and others, we deduce that for any fixed integer h, the total number of graphs on n vertices with obstacle number at most h is at most . This implies that there are bipartite graphs with arbitrarily large obstacle number, which answers a question of Alpert etal. (Discret Comput Geom doi: 10.1007/s00454-009-9233-8 , 2009
Lower bounds on the obstacle number of graphs
Given a graph , an {\em obstacle representation} of is a set of points
in the plane representing the vertices of , together with a set of connected
obstacles such that two vertices of are joined by an edge if and only if
the corresponding points can be connected by a segment which avoids all
obstacles. The {\em obstacle number} of is the minimum number of obstacles
in an obstacle representation of . It is shown that there are graphs on
vertices with obstacle number at least
Obstacle Numbers of Planar Graphs
Given finitely many connected polygonal obstacles in the
plane and a set of points in general position and not in any obstacle, the
{\em visibility graph} of with obstacles is the (geometric)
graph with vertex set , where two vertices are adjacent if the straight line
segment joining them intersects no obstacle. The obstacle number of a graph
is the smallest integer such that is the visibility graph of a set of
points with obstacles. If is planar, we define the planar obstacle
number of by further requiring that the visibility graph has no crossing
edges (hence that it is a planar geometric drawing of ). In this paper, we
prove that the maximum planar obstacle number of a planar graph of order is
, the maximum being attained (in particular) by maximal bipartite planar
graphs. This displays a significant difference with the standard obstacle
number, as we prove that the obstacle number of every bipartite planar graph
(and more generally in the class PURE-2-DIR of intersection graphs of straight
line segments in two directions) of order at least is .Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Graphs with Plane Outside-Obstacle Representations
An \emph{obstacle representation} of a graph consists of a set of polygonal
obstacles and a distinct point for each vertex such that two points see each
other if and only if the corresponding vertices are adjacent. Obstacle
representations are a recent generalization of classical polygon--vertex
visibility graphs, for which the characterization and recognition problems are
long-standing open questions.
In this paper, we study \emph{plane outside-obstacle representations}, where
all obstacles lie in the unbounded face of the representation and no two
visibility segments cross. We give a combinatorial characterization of the
biconnected graphs that admit such a representation. Based on this
characterization, we present a simple linear-time recognition algorithm for
these graphs. As a side result, we show that the plane vertex--polygon
visibility graphs are exactly the maximal outerplanar graphs and that every
chordal outerplanar graph has an outside-obstacle representation.Comment: 12 pages, 7 figure
Graphs with no grid obstacle representation
A graph
G
= (
V; E
)
admits a
grid obstacle representation
, if
there exist a subset
Ω
of the planar integer grid
Z
2
and an embed-
ding
f
:
V
!
Z
2
such that no vertex of
G
is mapped into a point
of
Ω
, and two vertices
u; v
2
V
are connected by an edge of
G
if
and only if there is a shortest path along the edges of
Z
2
that con-
nects
f
(
u
)
and
f
(
v
)
and avoids all other elements of
Ω
[
f
(
V
)
. We
answer a question of Bishnu, Ghosh, Mathew, Mishra, and Paul,
by showing that there exist graphs that do not admit a grid obstacle
representation
On obstacle numbers
The obstacle number is a new graph parameter introduced by Alpert, Koch, and Laison (2010). Mukkamala et al. (2012) show that there exist graphs with n vertices having obstacle number in Ω(n/ log n). In this note, we up this lower bound to Ω(n/(log log n)2). Our proof makes use of an upper bound of Mukkamala et al. on the number of graphs having obstacle number at most h in such a way that any subsequent improvements to their upper bound will improve our lower bound
Geometric Graph Theory and Wireless Sensor Networks
In this work, we apply geometric and combinatorial methods to explore a variety of problems motivated by wireless sensor networks. Imagine sensors capable of communicating along straight lines except through obstacles like buildings or barriers, such that the communication network topology of the sensors is their visibility graph. Using a standard distributed algorithm, the sensors can build common knowledge of their network topology.
We first study the following inverse visibility problem: What positions of sensors and obstacles define the computed visibility graph, with fewest obstacles? This is the problem of finding a minimum obstacle representation of a graph. This minimum number is the obstacle number of the graph. Using tools from extremal graph theory and discrete geometry, we obtain for every constant h that the number of n-vertex graphs that admit representations with h obstacles is 2o(n2). We improve this bound to show that graphs requiring Ω(n / log2 n) obstacles exist.
We also study restrictions to convex obstacles, and to obstacles that are line segments. For example, we show that every outerplanar graph admits a representation with five convex obstacles, and that allowing obstacles to intersect sometimes decreases their required number.
Finally, we study the corresponding problem for sensors equipped with GPS. Positional information allows sensors to establish common knowledge of their communication network geometry, hence we wish to compute a minimum obstacle representation of a given straight-line graph drawing. We prove that this problem is NP-complete, and provide a O(logOPT)-factor approximation algorithm by showing that the corresponding hypergraph family has bounded Vapnik-Chervonenkis dimension