27,596 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 curves that cross a fixed curve
We prove that for every integer , the class of intersection graphs
of curves in the plane each of which crosses a fixed curve in at least one and
at most points is -bounded. This is essentially the strongest
-boundedness result one can get for this kind of graph classes. As a
corollary, we prove that for any fixed integers and , every
-quasi-planar topological graph on vertices with any two edges crossing
at most times has edges.Comment: Small corrections, improved presentatio
Planar subgraphs without low-degree nodes
We study the following problem: given a geometric graph G and an integer k, determine if G has a planar spanning subgraph (with the original embedding and straight-line edges) such that all nodes have degree at least k. If G is a unit disk graph, the problem is trivial to solve for k = 1. We show that even the slightest deviation from the trivial case (e.g., quasi unit disk graphs or k = 1) leads to NP-hard problems.Peer reviewe
Unimodular lattice triangulations as small-world and scale-free random graphs
Real-world networks, e.g. the social relations or world-wide-web graphs,
exhibit both small-world and scale-free behaviour. We interpret lattice
triangulations as planar graphs by identifying triangulation vertices with
graph nodes and one-dimensional simplices with edges. Since these
triangulations are ergodic with respect to a certain Pachner flip, applying
different Monte-Carlo simulations enables us to calculate average properties of
random triangulations, as well as canonical ensemble averages using an energy
functional that is approximately the variance of the degree distribution. All
considered triangulations have clustering coefficients comparable with real
world graphs, for the canonical ensemble there are inverse temperatures with
small shortest path length independent of system size. Tuning the inverse
temperature to a quasi-critical value leads to an indication of scale-free
behaviour for degrees . Using triangulations as a random graph model
can improve the understanding of real-world networks, especially if the actual
distance of the embedded nodes becomes important.Comment: 17 pages, 6 figures, will appear in New J. Phy
Coloring curves that cross a fixed curve
We prove that for every integer t\geqslant 1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most t points is -bounded. This is essentially the strongest -boundedness result one can get for those kind of graph classes. As a corollary, we prove that for any fixed integers 2 and 1, every k-quasi-planar topological graph on n vertices with any two edges crossing at most t times has edges
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