Rebuilding convex sets in graphs

Abstract

The usual distance between pairs of vertices in a graph naturally gives rise to the notion of an interval between a pair of vertices in a graph. This in turn allows us to extend the notions of convex sets, convex hull, and extreme points in Euclidean space to the vertex set of a graph. The extreme vertices of a graph are known to be precisely the simplicial vertices, i.e., the vertices whose neighborhoods are complete graphs. It is known that the class of graphs with the Minkowski-Krein-Milman property, i.e., the property that every convex set is the convex hull of its extreme points, is precisely the class of chordal graphs without induced 3-fans. We define a vertex to be a contour vertex if the eccentricity of every neighbor is at most as large as that of the vertex. In this paper we show that every convex set of vertices in a graph is the convex hull of the collection of its contour vertices. We characterize those graphs for which every convex set has the property that its contour vertices coincide with its extreme points. A set of vertices in a graph is a geodetic set if the union of the intervals between pairs of vertices in the set, taken over all pairs in the set, is the entire vertex set. We show that the contour vertices in distance hereditary graphs form a geodetic set.

Geodeticity of the contour of chordal graphs

“…Concerning the first mentioned extension of the Krein-Milman property, Cáceres et al [3] obtained a similar property valid for every graph, by considering, instead of the extreme vertices, the so-called contour vertices. As for the second generalization, consisting of using the geodetic closure operator, a number of results have recently been obtained [2,3,7,8].…”

“…As for the second generalization, consisting of using the geodetic closure operator, a number of results have recently been obtained [2,3,7,8]. For example, it has been proved that in the class of distance-hereditary graphs, every convex set is the geodetic closure of its contour vertices [3,8].…”

Convex and subharmonic functions on graphs

“…With this, the notions of convex and convex functions extend naturally to all graphs, see [2,3,10,12].…”

“…length of the shorted path between two vertices, which leads to one notion of convexity on graphs studied in [10,12]. The notion of d-convexity on graphs when d is the standard graph metric is equivalent to the more common notion of geodesic convexity [2,3].…”

“…Common to [2,3,10,12], one starts with a graph and then puts a convexity theory on it, by using the graph metric. However in Section 3 we show that convex sets and functions defined on graphs with respect to the graph metric have a few pleasant but mostly a large number of undesirable properties.…”