173 research outputs found
Simple Topological Drawings of -Planar Graphs
Every finite graph admits a \emph{simple (topological) drawing}, that is, a
drawing where every pair of edges intersects in at most one point. However, in
combination with other restrictions simple drawings do not universally exist.
For instance, \emph{-planar graphs} are those graphs that can be drawn so
that every edge has at most crossings (i.e., they admit a \emph{-plane
drawing}). It is known that for , every -planar graph admits a
-plane simple drawing. But for , there exist -planar graphs that
do not admit a -plane simple drawing. Answering a question by Schaefer, we
show that there exists a function such
that every -planar graph admits an -plane simple drawing, for all
. Note that the function depends on only and is
independent of the size of the graph. Furthermore, we develop an algorithm to
show that every -planar graph admits an -plane simple drawing.Comment: Appears in the Proceedings of the 28th International Symposium on
Graph Drawing and Network Visualization (GD 2020
On Optimal 2- and 3-Planar Graphs
A graph is -planar if it can be drawn in the plane such that no edge is
crossed more than times. While for , optimal -planar graphs, i.e.,
those with vertices and exactly edges, have been completely
characterized, this has not been the case for . For and ,
upper bounds on the edge density have been developed for the case of simple
graphs by Pach and T\'oth, Pach et al. and Ackerman, which have been used to
improve the well-known "Crossing Lemma". Recently, we proved that these bounds
also apply to non-simple - and -planar graphs without homotopic parallel
edges and self-loops.
In this paper, we completely characterize optimal - and -planar graphs,
i.e., those that achieve the aforementioned upper bounds. We prove that they
have a remarkably simple regular structure, although they might be non-simple.
The new characterization allows us to develop notable insights concerning new
inclusion relationships with other graph classes
Edge Partitions of Optimal -plane and -plane Graphs
A topological graph is a graph drawn in the plane. A topological graph is
-plane, , if each edge is crossed at most times. We study the
problem of partitioning the edges of a -plane graph such that each partite
set forms a graph with a simpler structure. While this problem has been studied
for , we focus on optimal -plane and -plane graphs, which are
-plane and -plane graphs with maximum density. We prove the following
results. (i) It is not possible to partition the edges of a simple optimal
-plane graph into a -plane graph and a forest, while (ii) an edge
partition formed by a -plane graph and two plane forests always exists and
can be computed in linear time. (iii) We describe efficient algorithms to
partition the edges of a simple optimal -plane graph into a -plane graph
and a plane graph with maximum vertex degree , or with maximum vertex
degree if the optimal -plane graph is such that its crossing-free edges
form a graph with no separating triangles. (iv) We exhibit an infinite family
of simple optimal -plane graphs such that in any edge partition composed of
a -plane graph and a plane graph, the plane graph has maximum vertex degree
at least and the -plane graph has maximum vertex degree at least .
(v) We show that every optimal -plane graph whose crossing-free edges form a
biconnected graph can be decomposed, in linear time, into a -plane graph and
two plane forests
On the Number of Edges of Fan-Crossing Free Graphs
A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1
if there are no k+1 edges , such that have a
common endpoint and crosses all . We prove a tight bound of 4n-8 on
the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9
bound for a straight-edge drawing. For k > 2, we prove an upper bound of
3(k-1)(n-2) edges. We also discuss generalizations to monotone graph
properties
Note on k-planar crossing numbers
The crossing number CR(G) of a graph G=(V,E) is the smallest number of edge crossings over all drawings of G in the plane. For any k≥1, the k-planar crossing number of G, CRk(G), is defined as the minimum of CR(G0)+CR(G1)+…+CR(Gk−1) over all graphs G0,G1,…,Gk−1 with ∪i=0 k−1Gi=G. It is shown that for every k≥1, we have CRk(G)≤([Formula presented]−[Formula presented])CR(G). This bound does not remain true if we replace the constant [Formula presented]−[Formula presented] by any number smaller than [Formula presented]. Some of the results extend to the rectilinear variants of the k-planar crossing number. © 2017 Elsevier B.V
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