# Minimal trees and monophonic convexity

• Volume: 32, Issue: 4, page 685-704
• ISSN: 2083-5892

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## Abstract

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Let V be a finite set and 𝓜 a collection of subsets of V. Then 𝓜 is an alignment of V if and only if 𝓜 is closed under taking intersections and contains both V and the empty set. If 𝓜 is an alignment of V, then the elements of 𝓜 are called convex sets and the pair (V,𝓜 ) is called an alignment or a convexity. If S ⊆ V, then the convex hull of S is the smallest convex set that contains S. Suppose X ∈ ℳ. Then x ∈ X is an extreme point for X if X∖{x} ∈ ℳ. A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. Let G = (V,E) be a connected graph and U a set of vertices of G. A subgraph T of G containing U is a minimal U-tree if T is a tree and if every vertex of V(T)∖U is a cut-vertex of the subgraph induced by V(T). The monophonic interval of U is the collection of all vertices of G that belong to some minimal U-tree. Several graph convexities are defined using minimal U-trees and structural characterizations of graph classes for which the corresponding collection of convex sets is a convex geometry are characterized.

## How to cite

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Jose Cáceres, Ortrud R. Oellermann, and M. L. Puertas. "Minimal trees and monophonic convexity." Discussiones Mathematicae Graph Theory 32.4 (2012): 685-704. <http://eudml.org/doc/271032>.

@article{JoseCáceres2012,
abstract = {Let V be a finite set and 𝓜 a collection of subsets of V. Then 𝓜 is an alignment of V if and only if 𝓜 is closed under taking intersections and contains both V and the empty set. If 𝓜 is an alignment of V, then the elements of 𝓜 are called convex sets and the pair (V,𝓜 ) is called an alignment or a convexity. If S ⊆ V, then the convex hull of S is the smallest convex set that contains S. Suppose X ∈ ℳ. Then x ∈ X is an extreme point for X if X∖\{x\} ∈ ℳ. A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. Let G = (V,E) be a connected graph and U a set of vertices of G. A subgraph T of G containing U is a minimal U-tree if T is a tree and if every vertex of V(T)∖U is a cut-vertex of the subgraph induced by V(T). The monophonic interval of U is the collection of all vertices of G that belong to some minimal U-tree. Several graph convexities are defined using minimal U-trees and structural characterizations of graph classes for which the corresponding collection of convex sets is a convex geometry are characterized.},
author = {Jose Cáceres, Ortrud R. Oellermann, M. L. Puertas},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {minimal trees; monophonic intervals of sets; k-monophonic convexity; convex geometries; -monophonic convexity},
language = {eng},
number = {4},
pages = {685-704},
title = {Minimal trees and monophonic convexity},
url = {http://eudml.org/doc/271032},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Jose Cáceres
AU - Ortrud R. Oellermann
AU - M. L. Puertas
TI - Minimal trees and monophonic convexity
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 4
SP - 685
EP - 704
AB - Let V be a finite set and 𝓜 a collection of subsets of V. Then 𝓜 is an alignment of V if and only if 𝓜 is closed under taking intersections and contains both V and the empty set. If 𝓜 is an alignment of V, then the elements of 𝓜 are called convex sets and the pair (V,𝓜 ) is called an alignment or a convexity. If S ⊆ V, then the convex hull of S is the smallest convex set that contains S. Suppose X ∈ ℳ. Then x ∈ X is an extreme point for X if X∖{x} ∈ ℳ. A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. Let G = (V,E) be a connected graph and U a set of vertices of G. A subgraph T of G containing U is a minimal U-tree if T is a tree and if every vertex of V(T)∖U is a cut-vertex of the subgraph induced by V(T). The monophonic interval of U is the collection of all vertices of G that belong to some minimal U-tree. Several graph convexities are defined using minimal U-trees and structural characterizations of graph classes for which the corresponding collection of convex sets is a convex geometry are characterized.
LA - eng
KW - minimal trees; monophonic intervals of sets; k-monophonic convexity; convex geometries; -monophonic convexity
UR - http://eudml.org/doc/271032
ER -

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