140 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
Applications of Graph Embedding in Mesh Untangling
The subject of this thesis is mesh untangling through graph embedding, a method of laying out graphs on a planar surface, using an algorithm based on the work of Fruchterman and Reingold[1]. Meshes are a variety of graph used to represent surfaces with a wide number of applications, particularly in simulation and modelling. In the process of simulation, simulated forces can tangle the mesh through deformation and stress. The goal of this thesis was to create a tool to untangle structured meshes of complicated shapes and surfaces, including meshes with holes or concave sides. The goals of graph embedding, such as minimizing edge crossings align very well with the objectives of mesh untangling. I have designed and tested a tool which I named MUT (Mesh Untangling Tool) on meshes of various types including triangular, polygonal, and hybrid meshes.
Previous methods of mesh untangling have largely been numeric or optimizationbased. Additionally, most untangling methods produce low quality graphs which must be smoothed separately to produce good meshes. Currently graph embedding techniques have only been used for smoothing of untangled meshes. I have developed a tool based on the Fruchterman-Reingold algorithm for force-directed layout[1] that effectively untangles and smooths meshes simultaneously using graph embedding techniques. It can untangle complicated meshes with irregular polygonal frames, internal holes, and other complications that previous methods struggle with. The MUT does this by using several different approaches: untangling the mesh in stages from the frame in and anchoring the mesh at corner points to stabilize the untangling
Untangling Circular Drawings: Algorithms and Complexity
We consider the problem of untangling a given (non-planar) straight-line
circular drawing of an outerplanar graph into a planar
straight-line circular drawing by shifting a minimum number of vertices to a
new position on the circle. For an outerplanar graph , it is clear that such
a crossing-free circular drawing always exists and we define the circular
shifting number shift as the minimum number of vertices that are
required to be shifted in order to resolve all crossings of . We show
that the problem Circular Untangling, asking whether shift
for a given integer , is NP-complete. For -vertex outerplanar graphs, we
obtain a tight upper bound of shift. Based on these results we study Circular Untangling for almost-planar
circular drawings, in which a single edge is involved in all the crossings. In
this case, we provide a tight upper bound shift and present a constructive polynomial-time algorithm to
compute the circular shifting number of almost-planar drawings.Comment: 20 pages, 10 figures, extended version of ISAAC 2021 pape
Separating sets of strings by finding matching patterns is almost always hard
© 2017 Elsevier B.V. We study the complexity of the problem of searching for a set of patterns that separate two given sets of strings. This problem has applications in a wide variety of areas, most notably in data mining, computational biology, and in understanding the complexity of genetic algorithms. We show that the basic problem of finding a small set of patterns that match one set of strings but do not match any string in a second set is difficult (NP-complete, W[2]-hard when parameterized by the size of the pattern set, and APX-hard). We then perform a detailed parameterized analysis of the problem, separating tractable and intractable variants. In particular we show that parameterizing by the size of pattern set and the number of strings, and the size of the alphabet and the number of strings give FPT results, amongst others
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry states that every planar graph can be
represented by an embedding in which every edge is represented by a straight
line segment. We consider generalizations of F\'{a}ry's theorem to surfaces
equipped with Riemannian metrics. In this setting, we require that every edge
is drawn as a shortest path between its two endpoints and we call an embedding
with this property a shortest path embedding. The main question addressed in
this paper is whether given a closed surface S, there exists a Riemannian
metric for which every topologically embeddable graph admits a shortest path
embedding. This question is also motivated by various problems regarding
crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have
this property. We provide flat metrics on the torus and the Klein bottle which
also have this property.
Then we show that for the unit square flat metric on the Klein bottle there
exists a graph without shortest path embeddings. We show, moreover, that for
large g, there exist graphs G embeddable into the orientable surface of genus
g, such that with large probability a random hyperbolic metric does not admit a
shortest path embedding of G, where the probability measure is proportional to
the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of
genus g, such that every graph embeddable into S can be embedded so that every
edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer
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