10,577 research outputs found
Planar embeddability of the vertices of a graph using a fixed point set is NP-hard
Let G = (V, E) be a graph with n vertices and let P be a set of n points in the plane. We show that deciding whether there is a planar straight-line embedding of G such that the vertices V are embedded onto the points P
is NP-complete, even when G is 2-connected and 2-outerplanar. This settles an open problem posed in [P. Bose. On embedding an outer-planar graph in a point set. Comput. Geom. Theory Appl., 23:303-312, November 2002. A preliminary version appeared in Graph Drawing (Proc. GD ’97), LNCS 1353, pg. 25-36, F. Brandenberg, D. Eppstein, M.T. Goodrich, S.G. Kobourov, G. Liotta, and P. Mutzel. Selected open problems in graph drawing. In Graph Drawing (Proc. GD’03), LNCS, 2003. To appear y M. Kaufmann and R. Wiese. Embedding vertices at points: Few bends suffice for planar graphs. Journal of Graph Algorithms and Applications, 6(1):115–129, 2002. A preliminary version appeared in Graph Drawing (Proc. GD ’99), LNCS 1731, pg. 165–174].Cornelis Lely Stichtin
On Universal Point Sets for Planar Graphs
A set P of points in R^2 is n-universal, if every planar graph on n vertices
admits a plane straight-line embedding on P. Answering a question by Kobourov,
we show that there is no n-universal point set of size n, for any n>=15.
Conversely, we use a computer program to show that there exist universal point
sets for all n<=10 and to enumerate all corresponding order types. Finally, we
describe a collection G of 7'393 planar graphs on 35 vertices that do not admit
a simultaneous geometric embedding without mapping, that is, no set of 35
points in the plane supports a plane straight-line embedding of all graphs in
G.Comment: Fixed incorrect numbers of universal point sets in the last par
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
Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends
We study the following classes of beyond-planar graphs: 1-planar, IC-planar,
and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar,
and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every
edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs
of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two
pairs of crossing edges share two vertices. We study the relations of these
beyond-planar graph classes (beyond-planar graphs is a collective term for the
primary attempts to generalize the planar graphs) to right-angle crossing (RAC)
graphs that admit compact drawings on the grid with few bends. We present four
drawing algorithms that preserve the given embeddings. First, we show that
every -vertex NIC-planar graph admits a NIC-planar RAC drawing with at most
one bend per edge on a grid of size . Then, we show that
every -vertex 1-planar graph admits a 1-planar RAC drawing with at most two
bends per edge on a grid of size . Finally, we make two
known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at
most one bend per edge and for drawing IC-planar RAC graphs straight-line
Monotone Grid Drawings of Planar Graphs
A monotone drawing of a planar graph is a planar straight-line drawing of
where a monotone path exists between every pair of vertices of in some
direction. Recently monotone drawings of planar graphs have been proposed as a
new standard for visualizing graphs. A monotone drawing of a planar graph is a
monotone grid drawing if every vertex in the drawing is drawn on a grid point.
In this paper we study monotone grid drawings of planar graphs in a variable
embedding setting. We show that every connected planar graph of vertices
has a monotone grid drawing on a grid of size , and such a
drawing can be found in O(n) time
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