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{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1112,
author = {Gary Chartrand and Frank Harary and Ping Zhang},
title = {Geodetic sets in graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {20},
year = {2000},
pages = {129-138},
zbl = {0958.05044},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1112}
}
Gary Chartrand; Frank Harary; Ping Zhang. Geodetic sets in graphs. Discussiones Mathematicae Graph Theory, Tome 20 (2000) pp. 129-138. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1112/
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