6 research outputs found
On the Maximum Crossing Number
Research about crossings is typically about minimization. In this paper, we
consider \emph{maximizing} the number of crossings over all possible ways to
draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009]
conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a
drawing with vertices in convex position, that maximizes the number of edge
crossings. We disprove this conjecture by constructing a planar graph on twelve
vertices that allows a non-convex drawing with more crossings than any convex
one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the
maximum number of crossings of a geometric graph and that the weighted
geometric case is NP-hard to approximate. We strengthen these results by
showing hardness of approximation even for the unweighted geometric case and
prove that the unweighted topological case is NP-hard.Comment: 16 pages, 5 figure
Bounds of the sum of edge lengths in linear arrangements of trees
A fundamental problem in network science is the normalization of the
topological or physical distance between vertices, that requires understanding
the range of variation of the unnormalized distances. Here we investigate the
limits of the variation of the physical distance in linear arrangements of the
vertices of trees. In particular, we investigate various problems on the sum of
edge lengths in trees of a fixed size: the minimum and the maximum value of the
sum for specific trees, the minimum and the maximum in classes of trees (bistar
trees and caterpillar trees) and finally the minimum and the maximum for any
tree. We establish some foundations for research on optimality scores for
spatial networks in one dimension.Comment: Title changed at proof stag
Edge crossings in random linear arrangements
In spatial networks vertices are arranged in some space and edges may cross.
When arranging vertices in a 1-dimensional lattice edges may cross when drawn
above the vertex sequence as it happens in linguistic and biological networks.
Here we investigate the general of problem of the distribution of edge
crossings in random arrangements of the vertices. We generalize the existing
formula for the expectation of this number in random linear arrangements of
trees to any network and derive an expression for the variance of the number of
crossings in an arbitrary layout relying on a novel characterization of the
algebraic structure of that variance in an arbitrary space. We provide compact
formulae for the expectation and the variance in complete graphs, complete
bipartite graphs, cycle graphs, one-regular graphs and various kinds of trees
(star trees, quasi-star trees and linear trees). In these networks, the scaling
of expectation and variance as a function of network size is asymptotically
power-law-like in random linear arrangements. Our work paves the way for
further research and applications in 1-dimension or investigating the
distribution of the number of crossings in lattices of higher dimension or
other embeddings.Comment: Generalised our theory from one-dimensional layouts to practically
any type of layout. This helps study the variance of the number of crossings
in graphs when their vertices are arranged on the surface of a sphere, or on
the plane. Moreover, we also give closed formulae for this variance on
particular types of graphs in both linear arrangements and general layout
Recommended from our members
Discrete Geometry
A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry