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Growth constants of minorclosed classes of graphs
"... A minorclosed class of graphs is a set
..."
On the growth rate of minorclosed classes of graphs
, 2007
"... A minorclosed class of graphs is a set of labelled graphs which is closed under isomorphism and under taking minors. For a minorclosed class G, we let gn be the number of graphs in G which have n vertices. A recent result of Norine et al. [12] shows that for all minorclosed class G, there is a co ..."
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Cited by 2 (0 self)
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constant c such that gn ≤ c n n!. Our main results show that the growth rate of gn is far from arbitrary. For example, no minorclosed class G has gn = c n+o(n) n! with 0 < c < 1 or 1 < c < ξ ≈ 1.76.
Structure in minorclosed classes of matroids
 Surveys in Combinatorics 2013, London Math. Soc. Lecture Notes 409
, 2013
"... This paper gives an informal introduction to structure theory for minorclosed classes of matroids representable over a fixed finite field. The early sections describe some historical results that give evidence that welldefined structure exists for members of such classes. In later sections we desc ..."
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describe the fundamental classes and other features that necessarily appear in structure theory for minorclosed classes of matroids. We conclude with an informal statement of the structure theorem itself. This theorem generalises the Graph Minors Structure Theorem of Robertson and Seymour. 1
Densities of MinorClosed Graph Families
, 2010
"... We define the limiting density of a minorclosed family of simple graphs F to be the smallest number k such that every nvertex graph in F has at most kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting densities. This set of numbers is countable, wellordered, and closed; i ..."
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We define the limiting density of a minorclosed family of simple graphs F to be the smallest number k such that every nvertex graph in F has at most kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting densities. This set of numbers is countable, wellordered, and closed
Growth rates of minorclosed classes of matroids
, 2008
"... For a minorclosed class M of matroids, we let h(k) denote the maximum number of elements of a simple rankk matroid in M. We prove that, if M does not contain all simple rank2 matroids, then h(k) grows either linearly, quadratically, or exponentially. ..."
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Cited by 15 (10 self)
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For a minorclosed class M of matroids, we let h(k) denote the maximum number of elements of a simple rankk matroid in M. We prove that, if M does not contain all simple rank2 matroids, then h(k) grows either linearly, quadratically, or exponentially.
Community detection in graphs
, 2009
"... The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of th ..."
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Cited by 801 (1 self)
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The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices
Diameter and Treewidth in MinorClosed Graph Families
, 1999
"... It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a ..."
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Cited by 114 (3 self)
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minorclosed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to boundedgenus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph
Graphbased algorithms for Boolean function manipulation
 IEEE TRANSACTIONS ON COMPUTERS
, 1986
"... In this paper we present a new data structure for representing Boolean functions and an associated set of manipulation algorithms. Functions are represented by directed, acyclic graphs in a manner similar to the representations introduced by Lee [1] and Akers [2], but with further restrictions on th ..."
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Cited by 3499 (47 self)
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In this paper we present a new data structure for representing Boolean functions and an associated set of manipulation algorithms. Functions are represented by directed, acyclic graphs in a manner similar to the representations introduced by Lee [1] and Akers [2], but with further restrictions
Results 1  10
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1,219,806