Steiner distance and convexity in graphs

Authors:
J. Cáceres;A. Márquez;M. L. Puertas
Affiliations:
Department of Statistics and Applied Mathematics, University of Almería, Spain;Department of Applied Mathematics I, University of Sevilla, Spain;Department of Statistics and Applied Mathematics, University of Almería, Spain
Venue:
European Journal of Combinatorics
Year:
2008

Citing 4
Cited 1

Convexity in graphs and hypergraphs

SIAM Journal on Algebraic and Discrete Methods
Steiner distance stable graphs

Discrete Mathematics
Graph classes: a survey

Graph classes: a survey
Convexity and HHD-Free Graphs

SIAM Journal on Discrete Mathematics

Computing simple-path convex hulls in hypergraphs

Information Processing Letters

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Abstract

We use the Steiner distance to define a convexity in the vertex set of a graph, which has a nice behavior in the well-known class of HHD-free graphs. For this graph class, we prove that any Steiner tree of a vertex set is included into the geodesical convex hull of the set, which extends the well-known fact that the Euclidean convex hull contains at least one Steiner tree for any planar point set. We also characterize the graph class where Steiner convexity becomes a convex geometry, and provide a vertex set that allows us to rebuild any convex set, using convex hull operation, in any graph.