Decomposition of a hypergraph by partial-edge separators

Malvestuto, Francesco M.; Moscarini, Marina

doi:10.1016/s0304-3975(98)00128-5

Cited by 11 publications

(12 citation statements)

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“…An efficient algorithm for computing -convex hulls in hypergraphs was given in [12]; it makes use of the acyclic hypergraph whose edges are the maximal sets that are not separable by partial edges of ℋ [20]. e hypergraph is called the compact hypergraph of ℋ [7,18,20].…”

Section: -Convexity In Hypergraphsmentioning

confidence: 99%

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Decomposable Convexities in Graphs and Hypergraphs

Malvestuto

2013

ISRN Combinatorics

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Given a connected hypergraph ℋwith vertex set V, a convexity space on ℋ is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in ℋ. e members of are called convex sets. e convex hull of a subset X of V is the smallest convex set containing X. By a cluster of ℋ we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on ℋ is decomposable if it satis�es the following three axioms: (i) the maximal clusters of ℋ form an acyclic hypergraph, (ii) every maximal cluster of ℋ is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on ℋ is fully speci�ed by the maximal clusters of ℋ in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of ℋ and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of ℋ are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as "monophonic" and "canonical" convexities in hypergraphs and "all-paths" convexity in graphs).

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Section: -Convexity In Hypergraphsmentioning

confidence: 99%

“…e hypergraph is called the compact hypergraph of ℋ [7,18,20]. Let ℋ denote the compact hypergraph of ℋ.…”

Section: -Convexity In Hypergraphsmentioning

confidence: 99%

Decomposable Convexities in Graphs and Hypergraphs

Malvestuto

2013

ISRN Combinatorics

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“…A nonempty subset X of V (H) is a compact set [19,20] of H if every two vertices in X are connected and are separated by no partial edge of H . Note that a maximal compact set of H is either an edge of H or the union of three or more partial edges of H [19].…”

Section: The Compact Hypergraph Of a Hypergraphmentioning

confidence: 99%

“…Note that a maximal compact set of H is either an edge of H or the union of three or more partial edges of H [19]. The compact components of H are the subhypergraphs of H induced by maximal compact sets, and the compact hypergraph (or ''compaction'' [19,20]) of H is the (simple) hypergraph having as its edges the maximal compact sets of H . An edge of the compact hypergraph of H that is not an edge of H is called compound; thus, a compact component of H is a nontrivial hypergraph if and only if its vertex set is a compound edge of the compact hypergraph of H. Let H be a connected and simple hypergraph with m edges and n vertices.…”

Section: The Compact Hypergraph Of a Hypergraphmentioning

confidence: 99%

“…An edge of the compact hypergraph of H that is not an edge of H is called compound; thus, a compact component of H is a nontrivial hypergraph if and only if its vertex set is a compound edge of the compact hypergraph of H. Let H be a connected and simple hypergraph with m edges and n vertices. The number of compact components of H is not greater than (3m − 1)/2 so that the dimension of the compact hypergraph of H is O(mn) [19]; moreover, the compact hypergraph of H can be constructed in O(m 3 n) time [19,20]. Finally, the compact hypergraph of a hypergraph H enjoys the following two basic properties [19,20].…”

Section: The Compact Hypergraph Of a Hypergraphmentioning

confidence: 99%

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Canonical and monophonic convexities in hypergraphs

Malvestuto

2009

Discrete Mathematics

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Known properties of "canonical connections" from database theory and of "closed sets" from statistics implicitly define a hypergraph convexity, here called canonical convexity (c-convexity), and provide an efficient algorithm to compute c-convex hulls. We characterize the class of hypergraphs in which c-convexity enjoys the Minkowski-Krein-Milman property. Moreover, we compare c-convexity with the natural extension to hypergraphs of monophonic convexity (or m-convexity), and prove that: (1) m-convexity is coarser than c-convexity, (2) m-convexity and c-convexity are equivalent in conformal hypergraphs, and (3) m-convex hulls can be computed in the same efficient way as c-convex hulls. (C) 2009 Elsevier B.V. All rights reserved

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