For two vertices u and v of a graph G, the closed interval I[u,v] consists of u, v, and all vertices lying in some u-v geodesic in G. If S is a set of vertices of G, then I[S] is the union of all sets I[u,v] for u, v ∈ S. If I[S] = V(G), then S is a geodetic set for G. The geodetic number g(G) is the minimum cardinality of a geodetic set. A set S of vertices in a graph G is uniform if the distance between every two distinct vertices of S is the same fixed number. A geodetic set is essential if for every two distinct vertices u,v ∈ S, there exists a third vertex w of G that lies in some u-v geodesic but in no x-y geodesic for x, y ∈ S and x,y ≠ u,v. It is shown that for every integer k ≥ 2, there exists a connected graph G with g(G) = k which contains a uniform, essential minimum geodetic set. A minimal geodetic set S has no proper subset which is a geodetic set. The maximum cardinality of a minimal geodetic set is the upper geodetic number g⁺(G). It is shown that every two integers a and b with 2 ≤ a ≤ b are realizable as the geodetic and upper geodetic numbers, respectively, of some graph and when a b the minimum order of such a graph is b+2.
@article{DMGT_2000_20_1_a9,
author = {Chartrand, Gary and Harary, Frank and Zhang, Ping},
title = {Geodetic sets in graphs},
journal = {Discussiones Mathematicae. Graph Theory},
pages = {129--138},
year = {2000},
volume = {20},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/DMGT_2000_20_1_a9/}
}
TY - JOUR
AU - Chartrand, Gary
AU - Harary, Frank
AU - Zhang, Ping
TI - Geodetic sets in graphs
JO - Discussiones Mathematicae. Graph Theory
PY - 2000
SP - 129
EP - 138
VL - 20
IS - 1
UR - http://geodesic.mathdoc.fr/item/DMGT_2000_20_1_a9/
LA - en
ID - DMGT_2000_20_1_a9
ER -
%0 Journal Article
%A Chartrand, Gary
%A Harary, Frank
%A Zhang, Ping
%T Geodetic sets in graphs
%J Discussiones Mathematicae. Graph Theory
%D 2000
%P 129-138
%V 20
%N 1
%U http://geodesic.mathdoc.fr/item/DMGT_2000_20_1_a9/
%G en
%F DMGT_2000_20_1_a9
Chartrand, Gary; Harary, Frank; Zhang, Ping. Geodetic sets in graphs. Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 1, pp. 129-138. http://geodesic.mathdoc.fr/item/DMGT_2000_20_1_a9/
