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Edge Erasures and Chordal Graphs [article]

Jared Culbertson and Dan P. Guralnik and Peter F. Stiller
2018 arXiv   pre-print
We prove several results about 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.  ... 
arXiv:1706.04537v2 fatcat:tjwgixku2fh7hoinnmrzw4l4za

Edge erasures and chordal graphs

Jared Culbertson, Dan P. Guralnik, Peter F. Stiller
2021 Electronic Journal of Graph Theory and Applications  
We prove several results about 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.  ... 
doi:10.5614/ejgta.2021.9.2.13 fatcat:caesmmgu5zfw5kzguolhu5kxne

Extendable shellability for d-dimensional complexes on d+3 vertices [article]

Jared Culbertson, Anton Dochtermann, Dan P. Guralnik, Peter F. Stiller
2021 arXiv   pre-print
The proof involves considering the structure of 'exposed' 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.  ... 
arXiv:1908.07155v2 fatcat:htpofilfufcbpaz2enhmqe673i

Extendable Shellability for $d$-Dimensional Complexes on $d+3$ Vertices

Jared Culbertson, Anton Dochtermann, Dan P. Guralnik, Peter F. Stiller
2020 Electronic Journal of Combinatorics  
The proof involves considering the structure of 'exposed' 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.  ... 
doi:10.37236/9120 fatcat:bjgljaydzrgkxm5tfyhxonzube

Exposed circuits, linear quotients, and chordal clutters [article]

Anton Dochtermann
2021 arXiv   pre-print
In a recent preprint Culbertson, Guralnik, 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.  ... 
arXiv:1812.08128v3 fatcat:xanpoyb6hbec5o5elm2jt5s6vi

Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve [article]

Hiroyuki Suzuki
2014 arXiv   pre-print
that the loop 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.  ... 
arXiv:1408.0849v1 fatcat:hdglcbr5jjfttna5f53exfjxfy

Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve

Hiroyuki Suzuki
2014 Kodai Mathematical Journal  
Acknowledgments I am very thankful to my advisor, K.Uchiyama, for introducing me to this research topic, 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.  ... 
doi:10.2996/kmj/1404393889 fatcat:et4p7lagt5fipn5nxbfhqfvyp4

Neighborhood Expansion Grammars [chapter]

John L. Pfaltz
2000 Lecture Notes in Computer Science  
But, these gramamars are most e ective only to generate, 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.  ... 
doi:10.1007/978-3-540-46464-8_3 fatcat:yq2ukgppgvhnxcogepa65ypznm

Multiple SLE type scaling limits: from local to global [article]

Alex Karrila
2019 arXiv   pre-print
We consider collections of N 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  ... 
arXiv:1903.10354v1 fatcat:irho7y4fkjgilojcwwdnmi5pg4

Conformal invariance of planar loop-erased random walks and uniform spanning trees

Wendelin Werner, Oded Schramm, Gregory F. Lawler
2004 Annals of Probability  
In particular, the limit exists 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.  ... 
doi:10.1214/aop/1079021469 fatcat:lmhxn7hjxbfpnesgewaudm3fu4

Conformal Invariance Of Planar Loop-Erased Random Walks and Uniform Spanning Trees [chapter]

Gregory F. Lawler, Oded Schramm, Wendelin Werner
2011 Selected Works of Oded Schramm  
In particular, the limit exists 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.  ... 
doi:10.1007/978-1-4419-9675-6_30 fatcat:qvkzty2ehzd65hqndtcsy7optu

Boundary behaviour of RW's on planar graphs and convergence of LERW to chordal SLE_2 [article]

Kohei Uchiyama
2017 arXiv   pre-print
This paper concerns a random walk on a planar 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] .  ... 
arXiv:1705.03224v1 fatcat:yhxdvobfivgwbjev2wlptbmk3u

Conformally invariant scaling limits (an overview and a collection of problems) [article]

Oded Schramm
2006 arXiv   pre-print
Before we present the open problems, the definition of SLE will be motivated 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).  ... 
arXiv:math/0602151v2 fatcat:glk7pdoqxbg7jcnrupbvlc5fjm

Random planar curves and Schramm-Loewner evolutions [article]

Wendelin Werner
2003 arXiv   pre-print
We review some of the results that have been derived in the last years on conformal invariance, scaling limits 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).  ... 
arXiv:math/0303354v1 fatcat:uhkfpzchuvh3zjbjzhsejnatda

Topics in loop measures and the loop-erased walk [article]

Gregory F. Lawler
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 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 .  ... 
arXiv:1709.07531v1 fatcat:lnrax767ibgnfk67myyqtex744
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