6,175 research outputs found
Bar 1-Visibility Drawings of 1-Planar Graphs
A bar 1-visibility drawing of a graph is a drawing of where each
vertex is drawn as a horizontal line segment called a bar, each edge is drawn
as a vertical line segment where the vertical line segment representing an edge
must connect the horizontal line segments representing the end vertices and a
vertical line segment corresponding to an edge intersects at most one bar which
is not an end point of the edge. A graph is bar 1-visible if has a bar
1-visibility drawing. A graph is 1-planar if has a drawing in a
2-dimensional plane such that an edge crosses at most one other edge. In this
paper we give linear-time algorithms to find bar 1-visibility drawings of
diagonal grid graphs and maximal outer 1-planar graphs. We also show that
recursive quadrangle 1-planar graphs and pseudo double wheel 1-planar graphs
are bar 1-visible graphs.Comment: 15 pages, 9 figure
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
L-Visibility Drawings of IC-planar Graphs
An IC-plane graph is a topological graph where every edge is crossed at most
once and no two crossed edges share a vertex. We show that every IC-plane graph
has a visibility drawing where every vertex is an L-shape, and every edge is
either a horizontal or vertical segment. As a byproduct of our drawing
technique, we prove that an IC-plane graph has a RAC drawing in quadratic area
with at most two bends per edge
Bar 1-Visibility Graphs and their relation to other Nearly Planar Graphs
A graph is called a strong (resp. weak) bar 1-visibility graph if its
vertices can be represented as horizontal segments (bars) in the plane so that
its edges are all (resp. a subset of) the pairs of vertices whose bars have a
-thick vertical line connecting them that intersects at most one
other bar.
We explore the relation among weak (resp. strong) bar 1-visibility graphs and
other nearly planar graph classes. In particular, we study their relation to
1-planar graphs, which have a drawing with at most one crossing per edge;
quasi-planar graphs, which have a drawing with no three mutually crossing
edges; the squares of planar 1-flow networks, which are upward digraphs with
in- or out-degree at most one. Our main results are that 1-planar graphs and
the (undirected) squares of planar 1-flow networks are weak bar 1-visibility
graphs and that these are quasi-planar graphs
Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer
Let be an -node planar graph. In a visibility representation of ,
each node of is represented by a horizontal line segment such that the line
segments representing any two adjacent nodes of are vertically visible to
each other. In the present paper we give the best known compact visibility
representation of . Given a canonical ordering of the triangulated , our
algorithm draws the graph incrementally in a greedy manner. We show that one of
three canonical orderings obtained from Schnyder's realizer for the
triangulated yields a visibility representation of no wider than
. Our easy-to-implement O(n)-time algorithm bypasses the
complicated subroutines for four-connected components and four-block trees
required by the best previously known algorithm of Kant. Our result provides a
negative answer to Kant's open question about whether is a
worst-case lower bound on the required width. Also, if has no degree-three
(respectively, degree-five) internal node, then our visibility representation
for is no wider than (respectively, ).
Moreover, if is four-connected, then our visibility representation for
is no wider than , matching the best known result of Kant and He. As a
by-product, we obtain a much simpler proof for a corollary of Wagner's Theorem
on realizers, due to Bonichon, Sa\"{e}c, and Mosbah.Comment: 11 pages, 6 figures, the preliminary version of this paper is to
appear in Proceedings of the 20th Annual Symposium on Theoretical Aspects of
Computer Science (STACS), Berlin, Germany, 200
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
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
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