9,234 research outputs found
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
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
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
On the Maximum Crossing Number
Research about crossings is typically about minimization. In this paper, we
consider \emph{maximizing} the number of crossings over all possible ways to
draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009]
conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a
drawing with vertices in convex position, that maximizes the number of edge
crossings. We disprove this conjecture by constructing a planar graph on twelve
vertices that allows a non-convex drawing with more crossings than any convex
one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the
maximum number of crossings of a geometric graph and that the weighted
geometric case is NP-hard to approximate. We strengthen these results by
showing hardness of approximation even for the unweighted geometric case and
prove that the unweighted topological case is NP-hard.Comment: 16 pages, 5 figure
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