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630
Loopy belief propagation for approximate inference: An empirical study. In:
 Proceedings of Uncertainty in AI,
, 1999
"... Abstract Recently, researchers have demonstrated that "loopy belief propagation" the use of Pearl's polytree algorithm in a Bayesian network with loops can perform well in the context of errorcorrecting codes. The most dramatic instance of this is the near Shannonlimit performanc ..."
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Cited by 676 (15 self)
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likelihood weighting 3.1 The PYRAMID network All nodes were binary and the conditional probabilities were represented by tablesentries in the conditional probability tables (CPTs) were chosen uniformly in the range (0, 1]. 3.2 The toyQMR network All nodes were binary and the conditional probabilities
Dense Minors in Graphs of Large Girth
 Combinatorica
"... this paper is to reduce the upper bound for the required girth to the correct order of magnitude: Theorem 1. For any inte k,e ve graph G of girth g(G) > 6 log k +3and #(G) # 3 has a minor H with #(H) >k. The best lower bound we have found is 8 3 log c, but we note that existing conjectu ..."
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Cited by 8 (0 self)
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conjectures about cubic graphs of large girth would raise this to about 4 log . Since an average degree of at least cr # log r forces a K r minor [ 5, 8 ], Theorem 1 has the following consequence: Corollary 2.The ee a constant c # R such thate ea graph G of girth g(G) # 6 log r<F1
Graph minors. X. Obstructions to treedecomposition
 J. COMB. THEORY, SERIES B
, 1991
"... Roughly, a graph has small “treewidth” if it can be constructed by piecing small graphs together in a tree structure. Here we study the obstructions to the existence of such a tree structure. We find, for instance: (i) a minimax formula relating treewidth with the largest such obstructions (ii) an ..."
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Cited by 207 (10 self)
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) an association between such obstructions and large grid minors of the graph (iii) a “treedecomposition” of the graph into pieces corresponding with the obstructions. These results will be of use in later papers.
Minors in Graphs of Large Girth
 J. Combin. Theory B
, 1988
"... We show that for every odd integer g 5 there exists a constant c such that every graph of minimum degree r and girth at least g contains a minor of minimum degree at least cr . This is best possible up to the value of the constant c for g = 5; 7 and 11. More generally, a wellknown conjecture ..."
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Cited by 5 (0 self)
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We show that for every odd integer g 5 there exists a constant c such that every graph of minimum degree r and girth at least g contains a minor of minimum degree at least cr . This is best possible up to the value of the constant c for g = 5; 7 and 11. More generally, a wellknown
Constructions for Cubic Graphs With Large Girth
 Electronic Journal of Combinatorics
, 1998
"... The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a welldefined integer ¯ 0 (g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value ¯ 0 (g) is attained by a ..."
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Cited by 50 (1 self)
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by a graph whose girth is exactly g. The values of ¯ 0 (g) when 3 g 8 have been known for over thirty years. For these values of g each minimal graph is unique and, apart from the case g = 7, a simple lower bound is attained. This paper is mainly concerned with what happens when g 9, where
Topological Minors in Graphs of Large Girth
 J. Combin. Theory B
, 1988
"... We prove that every graph of minimum degree at least r and girth at least 186 contains a subdivision of K_r+1 and that for r ≥ 435 a girth of at least 15 suces. This implies that the conjecture of Hajós that every graph of chromatic number at least r contains a subdivision of K_r (which is fa ..."
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Cited by 5 (2 self)
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is false in general) is true for graphs of girth at least 186 (or 15 if r ≥ 436). More generally, we show that for every graph H of maximum degree Δ(H) ≥ 2, every graph G of minimum degree at least max{Δ(H), 3} and girth at least 166 ... contains a subdivision of H
Unavoidable Minors Of Graphs Of Large Type
, 1999
"... . In this paper, we study one measure of complexity of a graph, namely its type. The type of a graph G is defined to be the minimum number n such that there is a sequence of graphs G = G 0 , G1 , : : : , Gn , where G i is obtained by contracting one edge in or deleting one edge from each block of ..."
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Cited by 1 (0 self)
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of G i\Gamma1 , and where Gn is edgeless. We show that a 3connected graph has large type if and only if it has a minor isomorphic to a large fan. Furthermore, we show that if a graph has large type, then it has a minor isomorphic to a large fan or to a large member of one of two specified families
Graph Minor Theory
 BULLETIN (NEW SERIES) OF THE AMERICAN MATHEMATICAL SOCIETY
, 2005
"... A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a farreaching ..."
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Cited by 23 (0 self)
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reaching generalization of this, conjectured by Wagner: If a class of graphs is minorclosed (i.e., it is closed under deleting and contracting edges), then it can be characterized by a finite number of excluded minors. The proof of this conjecture is based on a very general theorem about the structure of large graphs
Small minors in dense graphs
 European J. Combin
"... Abstract. A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of the whole graph. More precisely, we describe funct ..."
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Cited by 3 (0 self)
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(t); in particular f(3) = 2 + ε and f(4) = 4 + ε. For t ≤ 4, we establish similar results for graphs embedded on surfaces, where the size of the Ktmodel is bounded (for fixed t). 1.
On the excluded minor structure theorem for graphs of large treewidth
, 2009
"... At the core of the RobertsonSeymour theory of graph minors lies a powerful structure theorem which captures, for any fixed graph H, the common structural features of all the graphs not containing H as a minor. Robertson and Seymour prove several versions of this theorem, each stressing some particu ..."
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Cited by 4 (1 self)
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At the core of the RobertsonSeymour theory of graph minors lies a powerful structure theorem which captures, for any fixed graph H, the common structural features of all the graphs not containing H as a minor. Robertson and Seymour prove several versions of this theorem, each stressing some
Results 1  10
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