3,934 research outputs found
On the Obfuscation Complexity of Planar Graphs
Being motivated by John Tantalo's Planarity Game, we consider straight line
plane drawings of a planar graph with edge crossings and wonder how
obfuscated such drawings can be. We define , the obfuscation complexity
of , to be the maximum number of edge crossings in a drawing of .
Relating to the distribution of vertex degrees in , we show an
efficient way of constructing a drawing of with at least edge
crossings. We prove bounds (\delta(G)^2/24-o(1))n^2 < \obf G <3 n^2 for an
-vertex planar graph with minimum vertex degree .
The shift complexity of , denoted by , is the minimum number of
vertex shifts sufficient to eliminate all edge crossings in an arbitrarily
obfuscated drawing of (after shifting a vertex, all incident edges are
supposed to be redrawn correspondingly). If , then
is linear in the number of vertices due to the known fact that the matching
number of is linear. However, in the case we notice that
can be linear even if the matching number is bounded. As for
computational complexity, we show that, given a drawing of a planar graph,
it is NP-hard to find an optimum sequence of shifts making crossing-free.Comment: 12 pages, 1 figure. The proof of Theorem 3 is simplified. An overview
of a related work is adde
On the optimality of the Arf invariant formula for graph polynomials
We prove optimality of the Arf invariant formula for the generating function
of even subgraphs, or, equivalently, the Ising partition function, of a graph.Comment: Advances in Mathematics, 201
Algorithms for Visualizing Phylogenetic Networks
We study the problem of visualizing phylogenetic networks, which are
extensions of the Tree of Life in biology. We use a space filling visualization
method, called DAGmaps, in order to obtain clear visualizations using limited
space. In this paper, we restrict our attention to galled trees and galled
networks and present linear time algorithms for visualizing them as DAGmaps.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Strictly convex drawings of planar graphs
Every three-connected planar graph with n vertices has a drawing on an O(n^2)
x O(n^2) grid in which all faces are strictly convex polygons. These drawings
are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids.
More generally, a strictly convex drawing exists on a grid of size O(W) x
O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds
are obtained when the faces have fewer sides.
In the proof, we derive an explicit lower bound on the number of primitive
vectors in a triangle.Comment: 20 pages, 13 figures. to be published in Documenta Mathematica. The
revision includes numerous small additions, corrections, and improvements, in
particular: - a discussion of the constants in the O-notation, after the
statement of thm.1. - a different set-up and clarification of the case
distinction for Lemma
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
Computing k-Modal Embeddings of Planar Digraphs
Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal, if every vertex of G is incident to at most k pairs of consecutive edges with opposite orientations, i.e., the incoming and the outgoing edges at each vertex are grouped by the embedding into at most k sets of consecutive edges with the same orientation. In this paper, we study the k-Modality problem, which asks for the existence of a k-modal embedding of a planar digraph. This combinatorial problem is at the very core of a variety of constrained embedding questions for planar digraphs and flat clustered networks.
First, since the 2-Modality problem can be easily solved in linear time, we consider the general k-Modality problem for any value of k>2 and show that the problem is NP-complete for planar digraphs of maximum degree Delta <= k+3. We relate its computational complexity to that of two notions of planarity for flat clustered networks: Planar Intersection-Link and Planar NodeTrix representations. This allows us to answer in the strongest possible way an open question by Di Giacomo [https://doi.org/10.1007/978-3-319-73915-1_37], concerning the complexity of constructing planar NodeTrix representations of flat clustered networks with small clusters, and to address a research question by Angelini et al. [https://doi.org/10.7155/jgaa.00437], concerning intersection-link representations based on geometric objects that determine complex arrangements. On the positive side, we provide a simple FPT algorithm for partial 2-trees of arbitrary degree, whose running time is exponential in k and linear in the input size. Second, motivated by the recently-introduced planar L-drawings of planar digraphs [https://doi.org/10.1007/978-3-319-73915-1_36], which require the computation of a 4-modal embedding, we focus our attention on k=4. On the algorithmic side, we show a complexity dichotomy for the 4-Modality problem with respect to Delta, by providing a linear-time algorithm for planar digraphs with Delta <= 6. This algorithmic result is based on decomposing the input digraph into its blocks via BC-trees and each of these blocks into its triconnected components via SPQR-trees. In particular, we are able to show that the constraints imposed on the embedding by the rigid triconnected components can be tackled by means of a small set of reduction rules and discover that the algorithmic core of the problem lies in special instances of NAESAT, which we prove to be always NAE-satisfiable - a result of independent interest that improves on Porschen et al. [https://doi.org/10.1007/978-3-540-24605-3_14]. Finally, on the combinatorial side, we consider outerplanar digraphs and show that any such a digraph always admits a k-modal embedding with k=4 and that this value of k is best possible for the digraphs in this family
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