62,880 research outputs found
Minimal Graphs with Crossing Number at Least k
AbstractIt is easily seen that. for each k, there is a graph G whose crossing number is at least k and an edge e of G such that G − e is planar. We show that there is a positive constant c such that, if G is a graph with crossing number k, then there exists an edge e of G such that G − e has crossing number at least ck
Geometric Crossing-Minimization - A Scalable Randomized Approach
We consider the minimization of edge-crossings in geometric drawings of graphs G=(V, E), i.e., in drawings where each edge is depicted as a line segment. The respective decision problem is NP-hard [Daniel Bienstock, 1991]. Crossing-minimization, in general, is a popular theoretical research topic; see Vrt\u27o [Imrich Vrt\u27o, 2014]. In contrast to theory and the topological setting, the geometric setting did not receive a lot of attention in practice. Prior work [Marcel Radermacher et al., 2018] is limited to the crossing-minimization in geometric graphs with less than 200 edges. The described heuristics base on the primitive operation of moving a single vertex v to its crossing-minimal position, i.e., the position in R^2 that minimizes the number of crossings on edges incident to v.
In this paper, we introduce a technique to speed-up the computation by a factor of 20. This is necessary but not sufficient to cope with graphs with a few thousand edges. In order to handle larger graphs, we drop the condition that each vertex v has to be moved to its crossing-minimal position and compute a position that is only optimal with respect to a small random subset of the edges. In our theoretical contribution, we consider drawings that contain for each edge uv in E and each position p in R^2 for v o(|E|) crossings. In this case, we prove that with a random subset of the edges of size Theta(k log k) the co-crossing number of a degree-k vertex v, i.e., the number of edge pairs uv in E, e in E that do not cross, can be approximated by an arbitrary but fixed factor delta with high probability. In our experimental evaluation, we show that the randomized approach reduces the number of crossings in graphs with up to 13 000 edges considerably. The evaluation suggests that depending on the degree-distribution different strategies result in the fewest number of crossings
Crossing numbers of composite knots and spatial graphs
We study the minimal crossing number of composite knots
, where and are prime, by relating it to the minimal
crossing number of spatial graphs, in particular the -theta curve
that results from tying of the edges of the planar
embedding of the -theta graph into and the remaining edges into
. We prove that for large enough we have
. We also formulate additional
relations between the crossing numbers of certain spatial graphs that, if
satisfied, imply the additivity of the crossing number or at least give a lower
bound for .Comment: 20 pages, 11 figures, changes from version1: added Lemma 5.2 and
corrected mistake in Proposition 5.3, improved quality of figure
Large Non-Planar Graphs and an Application to Crossing-Critical Graphs
We prove that, for every positive integer k, there is an integer N such that
every 4-connected non-planar graph with at least N vertices has a minor
isomorphic to K_{4,k}, the graph obtained from a cycle of length 2k+1 by adding
an edge joining every pair of vertices at distance exactly k, or the graph
obtained from a cycle of length k by adding two vertices adjacent to each other
and to every vertex on the cycle. We also prove a version of this for
subdivisions rather than minors, and relax the connectivity to allow 3-cuts
with one side planar and of bounded size. We deduce that for every integer k
there are only finitely many 3-connected 2-crossing-critical graphs with no
subdivision isomorphic to the graph obtained from a cycle of length 2k by
joining all pairs of diagonally opposite vertices.Comment: To appear in Journal of Combinatorial Theory B. 20 pages. No figures.
Te
On embeddings of CAT(0) cube complexes into products of trees
We prove that the contact graph of a 2-dimensional CAT(0) cube complex of maximum degree can be coloured with at most
colours, for a fixed constant . This implies
that (and the associated median graph) isometrically embeds in the
Cartesian product of at most trees, and that the event
structure whose domain is admits a nice labeling with
labels. On the other hand, we present an example of a
5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes
which cannot be embedded into a Cartesian product of a finite number of trees.
This answers in the negative a question raised independently by F. Haglund, G.
Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the
computation of the bounds in Theorem 1. Some figures repaire
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