5,465 research outputs found
Improved enumeration of simple topological graphs
A simple topological graph T = (V (T ), E(T )) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have
the same set of pairs of crossing edges. We generalize results of Pach and TĂłth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2
O(n2log(m/n)), and at most 2O(mn1/2 log n) if m ≤ n 3/2. As a consequence we obtain a new upper bound 2 O(n3/2 log n) on the number of intersection
graphs of n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with n vertices to 2n2 ·α(n) O(1), using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize G is at most 2 m2+O(mn) and at least 2
Ω(m2) for graphs with m > (6 + ε)n.Graph Drawings and Representations, EuroGIGA ProjectCentre Interfacultaire Bernoull
Fixed parameter tractable algorithms in combinatorial topology
To enumerate 3-manifold triangulations with a given property, one typically
begins with a set of potential face pairing graphs (also known as dual
1-skeletons), and then attempts to flesh each graph out into full
triangulations using an exponential-time enumeration. However, asymptotically
most graphs do not result in any 3-manifold triangulation, which leads to
significant "wasted time" in topological enumeration algorithms. Here we give a
new algorithm to determine whether a given face pairing graph supports any
3-manifold triangulation, and show this to be fixed parameter tractable in the
treewidth of the graph.
We extend this result to a "meta-theorem" by defining a broad class of
properties of triangulations, each with a corresponding fixed parameter
tractable existence algorithm. We explicitly implement this algorithm in the
most generic setting, and we identify heuristics that in practice are seen to
mitigate the large constants that so often occur in parameterised complexity,
highlighting the practicality of our techniques.Comment: 16 pages, 9 figure
Statistical mechanics of topological phase transitions in networks
We provide a phenomenological theory for topological transitions in
restructuring networks. In this statistical mechanical approach energy is
assigned to the different network topologies and temperature is used as a
quantity referring to the level of noise during the rewiring of the edges. The
associated microscopic dynamics satisfies the detailed balance condition and is
equivalent to a lattice gas model on the edge-dual graph of a fully connected
network. In our studies -- based on an exact enumeration method, Monte-Carlo
simulations, and theoretical considerations -- we find a rich variety of
topological phase transitions when the temperature is varied. These transitions
signal singular changes in the essential features of the global structure of
the network. Depending on the energy function chosen, the observed transitions
can be best monitored using the order parameters Phi_s=s_{max}/M, i.e., the
size of the largest connected component divided by the number of edges, or
Phi_k=k_{max}/M, the largest degree in the network divided by the number of
edges. If, for example the energy is chosen to be E=-s_{max}, the observed
transition is analogous to the percolation phase transition of random graphs.
For this choice of the energy, the phase-diagram in the [,T] plane is
constructed. Single vertex energies of the form
E=sum_i f(k_i), where k_i is the degree of vertex i, are also studied.
Depending on the form of f(k_i), first order and continuous phase transitions
can be observed. In case of f(k_i)=-(k_i+c)ln(k_i), the transition is
continuous, and at the critical temperature scale-free graphs can be recovered.Comment: 12 pages, 12 figures, minor changes, added a new refernce, to appear
in PR
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Synthesising Graphical Theories
In recent years, diagrammatic languages have been shown to be a powerful and
expressive tool for reasoning about physical, logical, and semantic processes
represented as morphisms in a monoidal category. In particular, categorical
quantum mechanics, or "Quantum Picturalism", aims to turn concrete features of
quantum theory into abstract structural properties, expressed in the form of
diagrammatic identities. One way we search for these properties is to start
with a concrete model (e.g. a set of linear maps or finite relations) and start
composing generators into diagrams and looking for graphical identities.
Naively, we could automate this procedure by enumerating all diagrams up to a
given size and check for equalities, but this is intractable in practice
because it produces far too many equations. Luckily, many of these identities
are not primitive, but rather derivable from simpler ones. In 2010, Johansson,
Dixon, and Bundy developed a technique called conjecture synthesis for
automatically generating conjectured term equations to feed into an inductive
theorem prover. In this extended abstract, we adapt this technique to
diagrammatic theories, expressed as graph rewrite systems, and demonstrate its
application by synthesising a graphical theory for studying entangled quantum
states.Comment: 10 pages, 22 figures. Shortened and one theorem adde
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
Developing a Mathematical Model for Bobbin Lace
Bobbin lace is a fibre art form in which intricate and delicate patterns are
created by braiding together many threads. An overview of how bobbin lace is
made is presented and illustrated with a simple, traditional bookmark design.
Research on the topology of textiles and braid theory form a base for the
current work and is briefly summarized. We define a new mathematical model that
supports the enumeration and generation of bobbin lace patterns using an
intelligent combinatorial search. Results of this new approach are presented
and, by comparison to existing bobbin lace patterns, it is demonstrated that
this model reveals new patterns that have never been seen before. Finally, we
apply our new patterns to an original bookmark design and propose future areas
for exploration.Comment: 20 pages, 18 figures, intended audience includes Artists as well as
Computer Scientists and Mathematician
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