30,007 research outputs found
Minimum cycle and homology bases of surface embedded graphs
We study the problems of finding a minimum cycle basis (a minimum weight set
of cycles that form a basis for the cycle space) and a minimum homology basis
(a minimum weight set of cycles that generates the -dimensional
()-homology classes) of an undirected graph embedded on a
surface. The problems are closely related, because the minimum cycle basis of a
graph contains its minimum homology basis, and the minimum homology basis of
the -skeleton of any graph is exactly its minimum cycle basis.
For the minimum cycle basis problem, we give a deterministic
-time algorithm for graphs embedded on an orientable
surface of genus . The best known existing algorithms for surface embedded
graphs are those for general graphs: an time Monte Carlo
algorithm and a deterministic time algorithm. For the
minimum homology basis problem, we give a deterministic -time algorithm for graphs embedded on an orientable or non-orientable
surface of genus with boundary components, assuming shortest paths are
unique, improving on existing algorithms for many values of and . The
assumption of unique shortest paths can be avoided with high probability using
randomization or deterministically by increasing the running time of the
homology basis algorithm by a factor of .Comment: A preliminary version of this work was presented at the 32nd Annual
International Symposium on Computational Geometr
Minimum Cycle Base of Graphs Identified by Two Planar Graphs
In this paper, we study the minimum cycle base of the planar graphs obtained
from two 2-connected planar graphs by identifying an edge (or a cycle) of one graph with the corresponding edge (or cycle) of another, related with map geometries, i.e., Smarandache 2-dimensional manifolds
A Constructive Characterisation of Circuits in the Simple (2,2)-sparsity Matroid
We provide a constructive characterisation of circuits in the simple
(2,2)-sparsity matroid. A circuit is a simple graph G=(V,E) with |E|=2|V|-1 and
the number of edges induced by any is at most 2|X|-2.
Insisting on simplicity results in the Henneberg operation being enough only
when the graph is sufficiently connected. Thus we introduce 3 different join
operations to complete the characterisation. Extensions are discussed to when
the sparsity matroid is connected and this is applied to the theory of
frameworks on surfaces to provide a conjectured characterisation of when
frameworks on an infinite circular cylinder are generically globally rigid.Comment: 22 pages, 6 figures. Changes to presentatio
Tutte's dichromate for signed graphs
We introduce the ``trivariate Tutte polynomial" of a signed graph as an
invariant of signed graphs up to vertex switching that contains among its
evaluations the number of proper colorings and the number of nowhere-zero
flows. In this, it parallels the Tutte polynomial of a graph, which contains
the chromatic polynomial and flow polynomial as specializations. The number of
nowhere-zero tensions (for signed graphs they are not simply related to proper
colorings as they are for graphs) is given in terms of evaluations of the
trivariate Tutte polynomial at two distinct points. Interestingly, the
bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share
many similar properties with the Tutte polynomial of a graph, does not in
general yield the number of nowhere-zero flows of a signed graph. Therefore the
``dichromate" for signed graphs (our trivariate Tutte polynomial) differs from
the dichromatic polynomial (the rank-size generating function).
The trivariate Tutte polynomial of a signed graph can be extended to an
invariant of ordered pairs of matroids on a common ground set -- for a signed
graph, the cycle matroid of its underlying graph and its frame matroid form the
relevant pair of matroids. This invariant is the canonically defined Tutte
polynomial of matroid pairs on a common ground set in the sense of a recent
paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and
Kayibi as a four-variable linking polynomial of a matroid pair on a common
ground set.Comment: 53 pp. 9 figure
Minimal length product over homology bases of manifolds
Minkowski's second theorem can be stated as an inequality for -dimensional
flat Finsler tori relating the volume and the minimal product of the lengths of
closed geodesics which form a homology basis. In this paper we show how this
fundamental result can be promoted to a principle holding for a larger class of
Finsler manifolds. This includes manifolds for which first Betti number and
dimension do no necessarily coincide, a prime example being the case of
surfaces. This class of manifolds is described by a non-vanishing condition for
the hyperdeterminant reduced modulo of the multilinear map induced by the
fundamental class of the manifold on its first -cohomology group
using the cup product.Comment: 24 page
Homological Error Correction: Classical and Quantum Codes
We prove several theorems characterizing the existence of homological error
correction codes both classically and quantumly. Not every classical code is
homological, but we find a family of classical homological codes saturating the
Hamming bound. In the quantum case, we show that for non-orientable surfaces it
is impossible to construct homological codes based on qudits of dimension
, while for orientable surfaces with boundaries it is possible to
construct them for arbitrary dimension . We give a method to obtain planar
homological codes based on the construction of quantum codes on compact
surfaces without boundaries. We show how the original Shor's 9-qubit code can
be visualized as a homological quantum code. We study the problem of
constructing quantum codes with optimal encoding rate. In the particular case
of toric codes we construct an optimal family and give an explicit proof of its
optimality. For homological quantum codes on surfaces of arbitrary genus we
also construct a family of codes asymptotically attaining the maximum possible
encoding rate. We provide the tools of homology group theory for graphs
embedded on surfaces in a self-contained manner.Comment: Revtex4 fil
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