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On a 2-factor with a specified edge in a graph satisfying the Ore condition

2002
*
Discrete Mathematics
*

*In*this article, we prove that if

*2*(G) ¿ |V (G)|, then for an

*edge*e of G, there exists

*a*

*2*-

*factor*

*with*k cycles that contains e,

*or*|V (G)| is even and G has

*a*vertex cover of size |V (G)|=

*2*containing ... Let k be an integer greater than

*one*, and let G be

*a*simple

*graph*

*with*at least 4k + 1 vertices. ... Acknowledgements

*The*authors wish to thank Dr. Y. Ishigami for his helpful advices and anonymous referees for appropriate comments. ...

##
###
Graphical models in Macaulay2
[article]

2013
*
arXiv
*
pre-print

*graphs*, and for checking for identifiable parameters

*in*Gaussian mixed

*graph*models. ...

*The*Macaulay2 package GraphicalModels contains algorithms for

*the*algebraic study of graphical models associated to undirected, directed and mixed

*graphs*, and associated collections of

*conditional*independence ... Work

*on*

*the*package was carried out during

*the*2010 and 2011 Macaulay2 workshops, which were partially supported by

*the*US National Science Foundation and

*the*Institute for Mathematics and Its Applications ...

##
###
Graphical models

2013
*
The Journal of Software for Algebra and Geometry
*

*graphs*, and for checking for identifiable parameters

*in*Gaussian mixed

*graph*models. ...

*The*Macaulay2 package GraphicalModels contains algorithms for

*the*algebraic study of graphical models associated to undirected, directed and mixed

*graphs*, and associated collections of

*conditional*independence ... There are two natural ways to

*specify*

*a*graphical model, through either

*conditional*independence statements

*specified*by

*the*

*graph*

*or*via

*a*parametric representation (often called

*a*"

*factorization*"). ...

##
###
A factoring algorithm using polygon-to-chain reductions for computing K-terminal network reliability

1985
*
Networks
*

Let G = (V,E) be an undirected

doi:10.1002/net.3230150204
fatcat:ph2ispuervdvtkpk2ngpgidafe
*graph*whose*edges*may fail, and let G, denote G*with**a*set K V*specified*.*Edge*failures are assumed to be statistically independent and to have known probabilities. ... Removing*the*restriction*on**edge*selection leaves results essentially unchanged*in**the*upper range of IK(, but minimum domination becomes only*a*tight upper bound for*the*lower range. analyze*the*complexity ... Then, under*the*same*conditions*we show that an optimal*edge*-selection strategy can be*specified*when*2*I IKI 5 151*or*IVI -*2*5 IKI 5 IVI. ...##
###
A look at cycles containing specified elements of a graph

2009
*
Discrete Mathematics
*

*edges*of

*specified*size,

*a*set of independent

*edges*

*satisfying*

*a*

*specified*property,

*a*set of disjoint paths containing

*a*fixed number of

*edges*,

*a*combination of vertices and

*edges*,

*or*

*a*linear forest ...

*One*possibility is to ask any of these questions

*in*

*a*fixed special class of

*graphs*like planar

*graphs*

*or*hypercubes. For lack of space, we shall not concentrate

*on*these questions here. ... Acknowledgements

*The*author would like to thank

*the*referees for their very helpful comments. ...

##
###
Compatible 2-factors

1992
*
Discrete Applied Mathematics
*

Given

doi:10.1016/0166-218x(92)90257-b
fatcat:6n556d5al5d7xf4amcjf4khn24
*a**graph*G and, at each vertex o,*a*set of pairs of*edges*incident to I, (called allowed transitions),*a**2*-*factor*Fof G is called compatible (*with**the*transition system) if for each vertex u,*the*... Such*a**2*-*factor*F is called compatibie*with*T= (T(U), v E V) (briefly*a*compatible*2*-*factor*). ... Acknowledgement*The*authors would like to thank*the*referees for valuable comments and suggestions, leading*in*particular to simplification of*the*proofs of Lemma 4.1 and Theorem 6. I. ...##
###
Knowledge representation and update in hierarchies of graphs
[article]

2020
*
arXiv
*
pre-print

*The*implementation of this theory,

*in*

*the*ReGraph Python library for (simple) directed

*graphs*

*with*attributes

*on*nodes and

*edges*, is then discussed

*in*

*the*context of two significant use cases. ...

*The*

*conditions*under which knowledge update,

*in*

*the*form of

*the*sesqui-pushout rewriting of an object

*in*

*a*hierarchy, can be propagated to

*the*rest of

*the*hierarchy,

*in*order to maintain all required path ... of

*the*hierarchy

*in*such

*a*way that

*the*commutativity

*condition*is

*satisfied*. ...

##
###
Low-dimensional embedding with extra information

2004
*
Proceedings of the twentieth annual symposium on Computational geometry - SCG '04
*

More precisely, we are given

doi:10.1145/997817.997866
dblp:conf/compgeom/BadoiuDHI04
fatcat:f7j6hlcsgbgvjc4yolpbwfx3ay
*a**graph*,*the*approximate lengths of*the**edges*, and possibly extra information, and our goal is to assign coordinates to*the*vertices that*satisfy**the*given constraints up to ...*a*constant*factor*away from*the*best possible. ... LEMMA 8 . 8 Given*a**graph*G*with**edge*lengths*specified*by D, we can check*in*polynomial time whether there is an embedding f that*satisfies**the*range*condition*and matches D exactly*on**the**edges*of f ...##
###
MAP Estimation Via Agreement on Trees: Message-Passing and Linear Programming

2005
*
IEEE Transactions on Information Theory
*

We develop and analyze methods for computing provably optimal maximum

doi:10.1109/tit.2005.856938
fatcat:uovgjk2a5bddtcuga2wo7erure
*a*posteriori probability (MAP) configurations for*a*subclass of Markov random fields defined*on**graphs**with*cycles. ...*In*this way, we establish*a*connection between*a*certain LP relaxation of*the*mode-finding problem and*a*reweighted form of*the*max-product (min-sum) message-passing algorithm. ... ACKNOWLEDGMENT*The*authors wish to thank Jon Feldman and David Karger for helpful discussions. ...##
###
Generic pole assignability, structurally constrained controllers and unimodular completion

2013
*
Linear Algebra and its Applications
*

J.W. van der Woude for discussions

doi:10.1016/j.laa.2013.10.004
fatcat:yjf6mlwi5fbdfe3xphrgxmve5a
*in**the*context of Theorem 2.7. We also thank Dr. Fan Yang for pointing corrections*in**a*preliminary version. ... matrix*satisfies*certain 'homogeneity*conditions*'*on*its invariant*factors*:*the**factors*are assumed to be monomials. ... to this*edge*is*a*polynomial of degree*one**or*more. ...##
###
The structure and existence of 2-factors in iterated line graphs

2007
*
Discussiones Mathematicae Graph Theory
*

Specifically, we give degree

doi:10.7151/dmgt.1376
fatcat:jaflokhiknfvzhkxm5tjfplhfa
*conditions**on*G that ensure L*2*(G) contains*a**2*-*factor**with*every possible number of cycles, and we give*a*sufficient*condition*for*the*existence of*a**2*-*factor**in*L*2*(G) ... We prove several results about*the*structure of*2*-*factors**in*iterated line*graphs*. ... We also give*a*sufficient*condition*for*the*existence of*a**2*-*factor**in*L*2*(G)*with*all cycle lengths*specified*. ...##
###
Factoring Algorithms for Computing K-Terminal Network Reliability

1986
*
IEEE Transactions on Reliability
*

math needed to use results: Same Results useful to: Reliability analysts and theoreticians Abstract-Let GK denote

doi:10.1109/tr.1986.4335431
fatcat:xkrrloaqdjeubowd65hjybjhz4
*a**graph*G whose*edges*can fail and*with**a*set K C V*specified*. ...*A**factoring*algorithm for computing network reliability recursively applies*the*formula R(GK) = piR(GK * eC) + qiR(GKe) where GK * ec is GK,*with**edge*ec contracted, GjKei is GK*with*e deleted andpi m ... R7 is general, and is*the*dual of*a*type 1 polygon-to-chain reduction*in**a*planar*graph**with*KI =*2*. R8 is trivially derived by*factoring**on**the**edge*ea, hence its name. ...##
###
Low-Dimensional Embedding with Extra Information

2006
*
Discrete & Computational Geometry
*

*or*additive) error

*on*

*the*resulting distances and other information is within

*a*constant

*factor*of

*the*best possible. ... For general

*graphs*,

*the*analogous problem is NP-hard even

*with*exact distances. ... Acknowledgments We thank

*the*anonymous referees for their helpful comments. ...

##
###
Proper path-factors and interval edge-coloring of (3,4)-biregular bigraphs
[article]

2007
*
arXiv
*
pre-print

An interval coloring of

arXiv:0704.2650v1
fatcat:dnotowwggnejhiyy3hqd2gol3e
*a**graph*G is*a*proper coloring of E(G) by positive integers such that*the*colors*on**the**edges*incident to any vertex are consecutive. ...*A*(3,4)-biregular bigraph is*a*bipartite*graph**in*which each vertex of*one*part has degree 3 and each vertex of*the*other has degree 4; it is unknown whether these all have interval colorings. ...*On**the*other hand, (3, 4) -biregular bigraphs*with*multiple*edges*need not have proper path-*factors*, even if they*satisfy*Pyatkin's*condition*. ...##
###
The Ponzano-Regge model and Reidemeister torsion
[article]

2006
*
arXiv
*
pre-print

*The*Ponzano-Regge model of three-dimensional quantum gravity is well-defined when

*the*observables

*satisfy*

*a*certain

*condition*involving

*the*twisted cohomology. ...

*In*this case,

*the*partition function is defined

*in*terms of

*the*Reidemeister torsion. Some consequences for

*the*special cases of planar

*graphs*and knots are given. ... To

*specify*an observable, we need

*a*

*graph*embedded

*in*M, and some data

*on*each

*edge*of

*the*

*graph*. More precisely, let Γ be

*a*connected subcomplex consisting of

*edges*and vertices of M. ...

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