05499518

a

05499518 Pelayo, Ignacio M. Generalizing the Krein-Milman property in graph convexity spaces: a short survey. Changat, Manoj (ed.) et al., Convexity in discrete structures. Joint proceedings of the international instructional workshop on convexity in discrete structures, Thiruvananthapuram, Kerala, India, March 22--April 2, 2006 and the international workshop on metric and convex graph theory, Barcelona, Spain, June 12--16, 2006. Mysore: Ramanujan Mathematical Society (ISBN 978-81-902545-5-7/hbk). Ramanujan Mathematical Society Lecture Notes Series 5, 131-142 (2008). 2008 Mysore: Ramanujan Mathematical Society EN boundary contour extreme vertex , geodetic set graph convexity space hull set Krein-Milman property monophonic set Steiner set Summary: A convexity on a non-empty finite set $V$ is a family ${\cal C}$ of subsets of $V$ (to be regarded as convex sets), which is closed under intersections and which contains both $V$ and the empty set. The pair $(V,{\cal C})$ is called a convexity space. The smallest convex set containing a set $A\subseteq V$ is said to be the convex hull of $A$. Given a convex set $W\subseteq V$, a vertex $v\in W$ is called an extreme vertex of $W$ if the set $W\setminus\{v\}$ is also convex. A convex geometry is a convexity space satisfying the so-called Krein-Milman property: Every convex set of $V$ is the convex hull of its extreme vertices. A Krein-Milman type property is any property similar to the previous one in which either the set of vertices (the extreme vertex set) or the operator (the convex hull one), or both, are replaced. In this paper, we survey the state-of-art of these type of properties for graph convexity spaces.