# Decomposability of Abstract and Path-Induced Convexities in Hypergraphs

Francesco Mario Malvestuto; Marina Moscarini

Discussiones Mathematicae Graph Theory (2015)

- Volume: 35, Issue: 3, page 493-515
- ISSN: 2083-5892

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topFrancesco Mario Malvestuto, and Marina Moscarini. "Decomposability of Abstract and Path-Induced Convexities in Hypergraphs." Discussiones Mathematicae Graph Theory 35.3 (2015): 493-515. <http://eudml.org/doc/271209>.

@article{FrancescoMarioMalvestuto2015,

abstract = {An abstract convexity space on a connected hypergraph H with vertex set V (H) is a family C of subsets of V (H) (to be called the convex sets of H) such that: (i) C contains the empty set and V (H), (ii) C is closed under intersection, and (iii) every set in C is connected in H. A convex set X of H is a minimal vertex convex separator of H if there exist two vertices of H that are separated by X and are not separated by any convex set that is a proper subset of X. A nonempty subset X of V (H) is a cluster of H if in H every two vertices in X are not separated by any convex set. The cluster hypergraph of H is the hypergraph with vertex set V (H) whose edges are the maximal clusters of H. A convexity space on H is called decomposable if it satisfies the following three properties: (C1) the cluster hypergraph of H is acyclic, (C2) every edge of the cluster hypergraph of H is convex, (C3) for every nonempty proper subset X of V (H), a vertex v does not belong to the convex hull of X if and only if v is separated from X in H by a convex cluster. It is known that the monophonic convexity (i.e., the convexity induced by the set of chordless paths) on a connected hypergraph is decomposable. In this paper we first provide two characterizations of decomposable convexities and then, after introducing the notion of a hereditary path family in a connected hypergraph H, we show that the convexity space on H induced by any hereditary path family containing all chordless paths (such as the families of simple paths and of all paths) is decomposable.},

author = {Francesco Mario Malvestuto, Marina Moscarini},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {convex hull; hypergraph convexity; path-induced convexity; con- vex geometry.; convex geometry},

language = {eng},

number = {3},

pages = {493-515},

title = {Decomposability of Abstract and Path-Induced Convexities in Hypergraphs},

url = {http://eudml.org/doc/271209},

volume = {35},

year = {2015},

}

TY - JOUR

AU - Francesco Mario Malvestuto

AU - Marina Moscarini

TI - Decomposability of Abstract and Path-Induced Convexities in Hypergraphs

JO - Discussiones Mathematicae Graph Theory

PY - 2015

VL - 35

IS - 3

SP - 493

EP - 515

AB - An abstract convexity space on a connected hypergraph H with vertex set V (H) is a family C of subsets of V (H) (to be called the convex sets of H) such that: (i) C contains the empty set and V (H), (ii) C is closed under intersection, and (iii) every set in C is connected in H. A convex set X of H is a minimal vertex convex separator of H if there exist two vertices of H that are separated by X and are not separated by any convex set that is a proper subset of X. A nonempty subset X of V (H) is a cluster of H if in H every two vertices in X are not separated by any convex set. The cluster hypergraph of H is the hypergraph with vertex set V (H) whose edges are the maximal clusters of H. A convexity space on H is called decomposable if it satisfies the following three properties: (C1) the cluster hypergraph of H is acyclic, (C2) every edge of the cluster hypergraph of H is convex, (C3) for every nonempty proper subset X of V (H), a vertex v does not belong to the convex hull of X if and only if v is separated from X in H by a convex cluster. It is known that the monophonic convexity (i.e., the convexity induced by the set of chordless paths) on a connected hypergraph is decomposable. In this paper we first provide two characterizations of decomposable convexities and then, after introducing the notion of a hereditary path family in a connected hypergraph H, we show that the convexity space on H induced by any hereditary path family containing all chordless paths (such as the families of simple paths and of all paths) is decomposable.

LA - eng

KW - convex hull; hypergraph convexity; path-induced convexity; con- vex geometry.; convex geometry

UR - http://eudml.org/doc/271209

ER -

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