14,813 research outputs found
The number of edges in k-quasi-planar graphs
A graph drawn in the plane is called k-quasi-planar if it does not contain k
pairwise crossing edges. It has been conjectured for a long time that for every
fixed k, the maximum number of edges of a k-quasi-planar graph with n vertices
is O(n). The best known upper bound is n(\log n)^{O(\log k)}. In the present
note, we improve this bound to (n\log n)2^{\alpha^{c_k}(n)} in the special case
where the graph is drawn in such a way that every pair of edges meet at most
once. Here \alpha(n) denotes the (extremely slowly growing) inverse of the
Ackermann function. We also make further progress on the conjecture for
k-quasi-planar graphs in which every edge is drawn as an x-monotone curve.
Extending some ideas of Valtr, we prove that the maximum number of edges of
such graphs is at most 2^{ck^6}n\log n.Comment: arXiv admin note: substantial text overlap with arXiv:1106.095
On the size of planarly connected crossing graphs
We prove that if an -vertex graph can be drawn in the plane such that
each pair of crossing edges is independent and there is a crossing-free edge
that connects their endpoints, then has edges. Graphs that admit
such drawings are related to quasi-planar graphs and to maximal -planar and
fan-planar graphs.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Beyond Outerplanarity
We study straight-line drawings of graphs where the vertices are placed in
convex position in the plane, i.e., convex drawings. We consider two families
of graph classes with nice convex drawings: outer -planar graphs, where each
edge is crossed by at most other edges; and, outer -quasi-planar graphs
where no edges can mutually cross. We show that the outer -planar graphs
are -degenerate, and consequently that every
outer -planar graph can be -colored, and this
bound is tight. We further show that every outer -planar graph has a
balanced separator of size . This implies that every outer -planar
graph has treewidth . For fixed , these small balanced separators
allow us to obtain a simple quasi-polynomial time algorithm to test whether a
given graph is outer -planar, i.e., none of these recognition problems are
NP-complete unless ETH fails. For the outer -quasi-planar graphs we prove
that, unlike other beyond-planar graph classes, every edge-maximal -vertex
outer -quasi planar graph has the same number of edges, namely . We also construct planar 3-trees that are not outer
-quasi-planar. Finally, we restrict outer -planar and outer
-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence
on the boundary is a cycle in the graph. For each , we express closed outer
-planarity and \emph{closed outer -quasi-planarity} in extended monadic
second-order logic. Thus, closed outer -planarity is linear-time testable by
Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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
Strip Planarity Testing of Embedded Planar Graphs
In this paper we introduce and study the strip planarity testing problem,
which takes as an input a planar graph and a function and asks whether a planar drawing of exists
such that each edge is monotone in the -direction and, for any
with , it holds . The problem has strong
relationships with some of the most deeply studied variants of the planarity
testing problem, such as clustered planarity, upward planarity, and level
planarity. We show that the problem is polynomial-time solvable if has a
fixed planar embedding.Comment: 24 pages, 12 figures, extended version of 'Strip Planarity Testing'
(21st International Symposium on Graph Drawing, 2013
Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count
We show that triangle-free penny graphs have degeneracy at most two, list
coloring number (choosability) at most three, diameter , and
at most edges.Comment: 10 pages, 2 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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