Discussiones Mathematicae Graph Theory 32(4) (2012) 685-704
doi: 10.7151/dmgt.1638

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Minimal Trees and Monophonic Convexity

Jose Cáceresa, Ortrud R. Oellermannb
and
M.L. Puertasa

a Department of Statistics and Applied Mathematics
University of Almeria, 04120, Almeria, Spain
b Department of Mathematics and Statistics., University of Winnipeg,
515 Portage Ave, Winnipeg, R3B 2E9, Canada

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.

Keywords: minimal trees, monophonic intervals of sets, k-monophonic convexity, convex geometries

2010 Mathematics Subject Classification: 05C75, 05C12, 05C17, 05C05.

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Received 11 July 2011
Revised 20 December 2011
Accepted 21 December 2011