89,571 research outputs found
Cylindrical Graph Construction (definition and basic properties)
In this article we introduce the {\it cylindrical construction} for graphs
and investigate its basic properties. We state a main result claiming a weak
tensor-like duality for this construction. Details of our motivations and
applications of the construction will appear elsewhere
Comparing fat graph models of moduli space
Godin introduced the categories of open closed fat graphs and
admissible fat graphs as models of the mapping class group of open
closed cobordism. We use the contractibility of the arc complex to give a new
proof of Godin's result that is a model of the mapping class group
of open-closed cobordisms. Similarly, Costello introduced a chain complex of
black and white graphs -Graphs, as a rational homological model of mapping
class groups. We use the result on admissible fat graphs to give a new integral
proof of Costellos's result that -Graphs is a homological model of mapping
class groups. The nature of this proof also provides a direct connection
between both models which were previously only known to be abstractly
equivalent. Furthermore, we endow Godin's model with a composition structure
which models composition of cobordisms along their boundary and we use the
connection between both models to give -Graphs a composition structure and
show that -Graphs are actually a model for the open-closed cobordism
category.Comment: 45 pages, 24 figure
HOMFLY-PT homology for general link diagrams and braidlike isotopy
Khovanov and Rozansky's categorification of the HOMFLY-PT polynomial is
invariant under braidlike isotopies for any link diagram and Markov moves for
braid closures. To define HOMFLY-PT homology, they required a link to be
presented as a braid closure, because they did not prove invariance under the
other oriented Reidemeister moves. In this text we prove that the Reidemeister
IIb move fails in HOMFLY-PT homology by using virtual crossing filtrations of
the author and Rozansky. The decategorification of HOMFLY-PT homology for
general link diagrams gives a deformed version of the HOMFLY-PT polynomial,
, which can be used to detect nonbraidlike isotopies. Finally, we
will use to prove that HOMFLY-PT homology is not an invariant of
virtual links, even when virtual links are presented as virtual braid closures.Comment: 26 pages, many TikZ figure
From Zwiebach invariants to Getzler relation
We introduce the notion of Zwiebach invariants that generalize Gromov-Witten
invariants and homotopical algebra structures. We outline the induction
procedure that induces the structure of Zwiebach on the subbicomplex, that
gives the structure of Gromov-Witten invariants on subbicomplex with zero
diffferentials. We propose to treat Hodge dGBV with 1/12 axiom as the simplest
set of Zwiebach invariants, and explicitely prove that it induces WDVV and
Getzler equations in genera 0 and 1 respectively.Comment: 35 page
Twisted duality for embedded graphs
We consider two operations on an edge of an embedded graph (or equivalently a
ribbon graph): giving a half-twist to the edge and taking the partial dual with
respect to the edge. These two operations give rise to an action of
S_3^{|E(G)|}, the ribbon group, on G. The action of the ribbon group on
embedded graphs extends the concepts of duality, partial duality and Petrie
duality. We show that this ribbon group action gives a complete
characterization of duality in that if G is any cellularly embedded graph with
medial graph G_m, then the orbit of G under the group action is precisely the
set of all graphs with medial graphs isomorphic (as abstract graphs) to G_m. We
provide characterizations of special sets of twisted duals, such as the partial
duals, of embedded graphs in terms of medial graphs and we show how different
kinds of graph isomorphism give rise to these various notions of duality. The
ribbon group action then leads to a deeper understanding of the properties of,
and relationships among, various graph polynomials via the generalized
transition polynomial which interacts naturally with the ribbon group action.Comment: V3 contains significant changes including new results and some
reorganizaton. To appear in Transactions of the AM
On the Complexity of Sampling Nodes Uniformly from a Graph
We study a number of graph exploration problems in the following natural
scenario: an algorithm starts exploring an undirected graph from some seed
node; the algorithm, for an arbitrary node that it is aware of, can ask an
oracle to return the set of the neighbors of . (In social network analysis,
a call to this oracle corresponds to downloading the profile page of user
in a social network.) The goal of the algorithm is to either learn something
(e.g., average degree) about the graph, or to return some random function of
the graph (e.g., a uniform-at-random node), while accessing/downloading as few
nodes of the graph as possible. Motivated by practical applications, we study
the complexities of a variety of problems in terms of the graph's mixing time
and average degree -- two measures that are believed to be quite small in
real-world social networks, and that have often been used in the applied
literature to bound the performance of online exploration algorithms. Our main
result is that the algorithm has to access nodes to obtain, with probability at
least , an -additive approximation of the average of a
bounded function on the nodes of a graph -- this lower bound matches the
performance of an algorithm that was proposed in the literature. We also give
tight bounds for the problem of returning a close-to-uniform-at-random node
from the graph. Finally, we give lower bounds for the problems of estimating
the average degree of the graph, and the number of nodes of the graph
Matching and MIS for Uniformly Sparse Graphs in the Low-Memory MPC Model
The Massively Parallel Computation (MPC) model serves as a common abstraction
of many modern large-scale parallel computation frameworks and has recently
gained a lot of importance, especially in the context of classic graph
problems. Unsatisfactorily, all current -round MPC
algorithms seem to get fundamentally stuck at the linear-memory barrier: their
efficiency crucially relies on each machine having space at least linear in the
number of nodes. As this might not only be prohibitively large, but also
allows for easy if not trivial solutions for sparse graphs, we are interested
in the low-memory MPC model, where the space per machine is restricted to be
strongly sublinear, that is, for any .
We devise a degree reduction technique that reduces maximal matching and
maximal independent set in graphs with arboricity to the
corresponding problems in graphs with maximum degree in
rounds. This gives rise to -round algorithms, where is the
-dependency in the round complexity of maximal matching and maximal
independent set in graphs with maximum degree . A concurrent work by
Ghaffari and Uitto shows that .
For graphs with arboricity , this almost
exponentially improves over Luby's -round PRAM algorithm [STOC'85,
JALG'86], and constitutes the first -round maximal
matching algorithm in the low-memory MPC model, thus breaking the linear-memory
barrier. Previously, the only known subpolylogarithmic algorithm, due to
Lattanzi et al. [SPAA'11], required strongly superlinear, that is,
, memory per machine
An exploration of two infinite families of snarks
Thesis (M.S.) University of Alaska Fairbanks, 2019In this paper, we generalize a single example of a snark that admits a drawing with even rotational symmetry into two infinite families using a voltage graph construction techniques derived from cyclic Pseudo-Loupekine snarks. We expose an enforced chirality in coloring the underlying 5-pole that generated the known example, and use this fact to show that the infinite families are in fact snarks. We explore the construction of these families in terms of the blowup construction. We show that a graph in either family with rotational symmetry of order m has automorphism group of order m2m⁺¹. The oddness of graphs in both families is determined exactly, and shown to increase linearly with the order of rotational symmetry.Chapter 1: Introduction -- 1.1 General Graph Theory -- Chapter 2: Introduction to Snarks -- 2.1 History -- 2.2 Motivation -- 2.3 Loupekine Snarks and k-poles -- 2.4 Conditions on Triviality -- Chapter 3: The Construction of Two Families of Snarks -- 3.1 Voltage Graphs and Lifts -- 3.2 The Family of Snarks, Fm -- 3.3 A Second Family of Snarks, Rm -- Chapter 4: Results -- 4.1 Proof that the graphs Fm and Rm are Snarks -- 4.2 Interpreting Fm and Rm as Blowup Graphs -- 4.3 Automorphism Group -- 4.4 Oddness -- Chapter 5: Conclusions and Open Questions -- References
Random intersection graphs with communities
Random intersection graphs model networks with communities, assuming an
underlying bipartite structure of groups and individuals, where these groups
may overlap. Group memberships are generated through the bipartite
configuration model. Conditionally on the group memberships, the classical
random intersection graph is obtained by connecting individuals when they are
together in at least one group. We generalize this definition, allowing for
arbitrary community structures within the groups.
In our new model, groups might overlap and they have their own internal
structure described by a graph, the classical setting corresponding to groups
being complete graphs. Our model turns out to be tractable. We analyze the
overlapping structure of the communities, derive the asymptotic degree
distribution and the local clustering coefficient. These proofs rely on local
weak convergence, which also implies that subgraph counts converge. We further
exploit the connection to the bipartite configuration model, for which we also
prove local weak convergence, and which is interesting in its own right.Comment: 34 pages, 12 figure
Graphical virtual links and a polynomial of signed cyclic graphs
For a signed cyclic graph G, we can construct a unique virtual link L by
taking the medial construction and convert 4-valent vertices of the medial
graph to crossings according to the signs. If a virtual link can occur in this
way then we say that the virtual link is graphical. In the article we shall
prove that a virtual link L is graphical if and only if it is checkerboard
colorable. On the other hand, we introduce a polynomial F[G] for signed cyclic
graphs, which is defined via a deletion-marking recursion. We shall establish
the relationship between F[G] of a signed cyclic graph G and the bracket
polynomial of one of the virtual link diagrams associated with G. Finally we
give a spanning subgraph expansion for F[G].Comment: 14 pages, 14 figures, LaTeX documen
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