16,964 research outputs found
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
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
Coloring Kk-free intersection graphs of geometric objects in the plane
AbstractThe intersection graph of a collection C of sets is the graph on the vertex set C, in which C1,C2∈C are joined by an edge if and only if C1∩C2≠0̸. Erdős conjectured that the chromatic number of triangle-free intersection graphs of n segments in the plane is bounded from above by a constant. Here we show that it is bounded by a polylogarithmic function of n, which is the first nontrivial bound for this problem. More generally, we prove that for any t and k, the chromatic number of every Kk-free intersection graph of n curves in the plane, every pair of which have at most t points in common, is at most (ctlognlogk)clogk, where c is an absolute constant and ct only depends on t. We establish analogous results for intersection graphs of convex sets, x-monotone curves, semialgebraic sets of constant description complexity, and sets that can be obtained as the union of a bounded number of sets homeomorphic to a disk.Using a mix of results on partially ordered sets and planar separators, for large k we improve the best known upper bound on the number of edges of a k-quasi-planar topological graph with n vertices, that is, a graph drawn in the plane with curvilinear edges, no k of which are pairwise crossing. As another application, we show that for every ε>0 and for every positive integer t, there exist δ>0 and a positive integer n0 such that every topological graph with n≥n0 vertices, at least n1+ε edges, and no pair of edges intersecting in more than t points, has at least nδ pairwise intersecting edges
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
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
On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs
Fan-planar graphs were recently introduced as a generalization of 1-planar
graphs. A graph is fan-planar if it can be embedded in the plane, such that
each edge that is crossed more than once, is crossed by a bundle of two or more
edges incident to a common vertex. A graph is outer-fan-planar if it has a
fan-planar embedding in which every vertex is on the outer face. If, in
addition, the insertion of an edge destroys its outer-fan-planarity, then it is
maximal outer-fan-planar. In this paper, we present a polynomial-time algorithm
to test whether a given graph is maximal outer-fan-planar. The algorithm can
also be employed to produce an outer-fan-planar embedding, if one exists. On
the negative side, we show that testing fan-planarity of a graph is NP-hard,
for the case where the rotation system (i.e., the cyclic order of the edges
around each vertex) is given
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