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Edge Erasures and Chordal Graphs
[article]

2018
*
arXiv
*
pre-print

We prove several results about

arXiv:1706.04537v2
fatcat:tjwgixku2fh7hoinnmrzw4l4za
*chordal**graphs**and*weighted*chordal**graphs*by focusing on exposed*edges*. These are*edges*that are properly contained in a single maximal complete subgraph. ... This leads to a characterization of*chordal**graphs*via deletions of a sequence of exposed*edges*from a complete*graph*. ... The authors gratefully acknowledge the support of Air Force Office of Science Research under the LRIR 15RYCOR153, MURI FA9550-10-1-0567*and*FA9550-11-10223 grants, respectively. ...##
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Edge erasures and chordal graphs

2021
*
Electronic Journal of Graph Theory and Applications
*

We prove several results about

doi:10.5614/ejgta.2021.9.2.13
fatcat:caesmmgu5zfw5kzguolhu5kxne
*chordal**graphs**and*weighted*chordal**graphs*by focusing on exposed*edges*. These are*edges*that are properly contained in a single maximal complete subgraph. ... This leads to a characterization of*chordal**graphs*via deletions of a sequence of exposed*edges*from a complete*graph*. ... Acknowledgement The authors gratefully acknowledge the support of Air Force Office of Science Research under the LRIR 15RYCOR153, MURI FA9550-10-1-0567*and*FA9550-11-10223 grants, respectively. ...##
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Extendable shellability for d-dimensional complexes on d+3 vertices
[article]

2021
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arXiv
*
pre-print

The proof involves considering the structure of 'exposed'

arXiv:1908.07155v2
fatcat:htpofilfufcbpaz2enhmqe673i
*edges*in*chordal**graphs*as well as a connection to linear quotients of quadratic monomial ideals. ... Suppose G is a*chordal**graph**and*xy ∈ ∂G is an exposed*edge*. ... In [7] it is shown that a*graph*G on vertex set [n] = {1, 2, . . . , n} is*chordal*if*and*only if G can be obtained from the complete*graph*K n via a sequence of*erasures*. ...##
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Extendable Shellability for $d$-Dimensional Complexes on $d+3$ Vertices

2020
*
Electronic Journal of Combinatorics
*

The proof involves considering the structure of 'exposed'

doi:10.37236/9120
fatcat:bjgljaydzrgkxm5tfyhxonzube
*edges*in*chordal**graphs*as well as a connection to linear quotients of quadratic monomial ideals. ... Dan Guralnik is grateful to Kod*Lab*and*all its members for his time there,*and*particularly to Daniel E. Koditschek for encouraging this collaboration. ... Suppose G is a*chordal**graph**and*xy ∈ ∂G is an exposed*edge*. ...##
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Exposed circuits, linear quotients, and chordal clutters
[article]

2021
*
arXiv
*
pre-print

In a recent preprint Culbertson, Guralnik,

arXiv:1812.08128v3
fatcat:xanpoyb6hbec5o5elm2jt5s6vi
*and*Stiller give a new characterization of*chordal**graphs*in terms of sequences of what they call '*edge*-*erasures*'. ... A*graph*G is said to be*chordal*if it has no induced cycles of length four or more. ... Macaulay2 [17] was used extensively to compute examples*and*we have included calculations in figures below. ...##
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Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve
[article]

2014
*
arXiv
*
pre-print

that the loop

arXiv:1408.0849v1
fatcat:hdglcbr5jjfttna5f53exfjxfy
*erasure*of the conditioned walk converges, as δ↓ 0, to the*chordal*SLE_2 that connects a*and*b in D, provided that an invariance principle is valid for both the random walk*and*the dual walk ... In this paper we consider the natural random walk on a planar*graph**and*scale it by a small positive number δ. ... prior to its publication*and*letting me include a result therein as Proposition 6.1. ...##
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Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve

2014
*
Kodai Mathematical Journal
*

Acknowledgments I am very thankful to my advisor, K.Uchiyama, for introducing me to this research topic,

doi:10.2996/kmj/1404393889
fatcat:et4p7lagt5fipn5nxbfhqfvyp4
*and*for his encouragements*and*many invaluable*and*helpful comments. ... I also would like to thank Tokyo Institute of Technology for providing me a wonderful environment for my life*and*study. Last I thank to my family for their encouragements*and*supports. ... Since G δ is planar*graph*, S x δ can not cross the*edge*e, so that it cannot behave as a Brownian path*and*the invariance principle fails to hold. ...##
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Neighborhood Expansion Grammars
[chapter]

2000
*
Lecture Notes in Computer Science
*

But, these gramamars are most e ective only to generate,

doi:10.1007/978-3-540-46464-8_3
fatcat:yq2ukgppgvhnxcogepa65ypznm
*and*parse, strings. ... For example, we show that restricted neighborhood expansion grammars capture the essence of nite state*and*context free phrase structure grammars. ... Figure 1 : A sequence of neighborhood expansions generating*chordal**graphs*clique),*and*because every*chordal**graph*must have at least two extreme points 8, 6], every*chordal**graph*can be so generated. ...##
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Multiple SLE type scaling limits: from local to global
[article]

2019
*
arXiv
*
pre-print

We consider collections of N

arXiv:1903.10354v1
fatcat:irho7y4fkjgilojcwwdnmi5pg4
*chordal*random curves obtained from a critical lattice model on a planar*graph*, in the limit when a fine-mesh*graph*approximates a simply-connected domain. ... These results essentially only take as input certain crossing conditions, very similar to those introduced by Kemppainen*and*Smirnov,*and*they allow the identification of scaling limits via the martingale ... ,e 2N ) that we are interested in are these conditional USTs,*and*the random*chordal*curves γ G;1 , . . . , γ G;N are the*chordal**graph*paths consisting of the odd*edges*e 1 , e 3 , . . . , e 2N −1*and*...##
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Conformal invariance of planar loop-erased random walks and uniform spanning trees

2004
*
Annals of Probability
*

In particular, the limit exists

doi:10.1214/aop/1079021469
fatcat:lmhxn7hjxbfpnesgewaudm3fu4
*and*is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists*and*is conformally invariant. ... The results*and*proofs are not restricted to a particular choice of lattice. ... Give n a spanning tree T of C , let T t de note the*graph*whose vertices are the ve rtices of c t*and*whose*edges*are those*edges*e t suc h th at e ¢. T. ...##
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Conformal Invariance Of Planar Loop-Erased Random Walks and Uniform Spanning Trees
[chapter]

2011
*
Selected Works of Oded Schramm
*

In particular, the limit exists

doi:10.1007/978-1-4419-9675-6_30
fatcat:qvkzty2ehzd65hqndtcsy7optu
*and*is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists*and*is conformally invariant. ... The results*and*proofs are not restricted to a particular choice of lattice. ... Give n a spanning tree T of C , let T t de note the*graph*whose vertices are the ve rtices of c t*and*whose*edges*are those*edges*e t suc h th at e ¢. T. ...##
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Boundary behaviour of RW's on planar graphs and convergence of LERW to chordal SLE_2
[article]

2017
*
arXiv
*
pre-print

This paper concerns a random walk on a planar

arXiv:1705.03224v1
fatcat:yhxdvobfivgwbjev2wlptbmk3u
*graph**and*presents certain estimates concerning the harmonic measures for the walk in a grid domain which estimates are useful for showing the convergence ... We assume that the walk started at a fixed vertex of the*graph*satisfies the invariance principle as in Yadin*and*Yehudayoff [16] in which the convergence of LERW to a radial SLE is established in this ... In this paper we are concerned with the*chordal*case of LERW on a planar*graph*as in [12] . ...##
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Conformally invariant scaling limits (an overview and a collection of problems)
[article]

2006
*
arXiv
*
pre-print

Before we present the open problems, the definition of SLE will be motivated

arXiv:math/0602151v2
fatcat:glk7pdoqxbg7jcnrupbvlc5fjm
*and*explained,*and*a brief sketch of recent results will be presented. ... There is a random*graph*dual to the tree, which consists of the dual*edges*perpendicular to primal*edges*not in the tree. ... The Fortuin-Kasteleyn [28] (FK) model (a.k.a. the random cluster model) is a probability measure on the collection of all subsets of the set of*edges*E of a finite*graph*G = (V, E). ...##
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Random planar curves and Schramm-Loewner evolutions
[article]

2003
*
arXiv
*
pre-print

We review some of the results that have been derived in the last years on conformal invariance, scaling limits

arXiv:math/0303354v1
fatcat:uhkfpzchuvh3zjbjzhsejnatda
*and*properties of some two-dimensional random curves. ... probabilities, critical exponents) using SLE, relate SLE to planar Brownian motions (i.e. the determination of the critical exponents), planar self-avoiding walks, critical percolation, loop-erased random walks*and*... Uniform spanning trees, Wilson's algorithm Suppose that a connected finite*graph*G = (V, E) is given (V is the set of vertices*and*E is the set of*edges*). ...##
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Topics in loop measures and the loop-erased walk
[article]

2017
*
arXiv
*
pre-print

These are notes based on a course that I gave at the University of Chicago in Fall 2016 on "Loop measures

arXiv:1709.07531v1
fatcat:lnrax767ibgnfk67myyqtex744
*and*the loop-erased random walk." ... Adding a self-*edge*Suppose that (27) holds for a*graph*G = (A, E) with A = {x 1 , . . . , x n }*and*consider a new graphG = (A,Ẽ) by adding one self-*edgeẽ*at x 1 . ... Take the loop-*erasure*LE(S j [0, T j ])*and*add those*edges**and*vertices to the tree to formT j . ...
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