For two vertices u and v of a graph G, the set I(u, v) consists of all vertices lying on 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. A set S is a geodetic set if I(S) = V(G). A minimum geodetic set is a geodetic set of minimum cardinality and this cardinality is the geodetic number g(G). A subset T of a minimum geodetic set S is called a forcing subset for S if S is the unique minimum geodetic set containing T. The forcing geodetic number of S is the minimum cardinality among the forcing subsets of S, and the forcing geodetic number f(G) of G is the minimum forcing geodetic number among all minimum geodetic sets of G. The forcing geodetic numbers of several classes of graphs are determined. For every graph G, f(G) ≤ g(G). It is shown that for all integers a, b with 0 ≤ a ≤ b, a connected graph G such that f(G) = a and g(G) = b exists if and only if (a,b) ∉ (1,1),(2,2).
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1084,
author = {Gary Chartrand and Ping Zhang},
title = {The forcing geodetic number of a graph},
journal = {Discussiones Mathematicae Graph Theory},
volume = {19},
year = {1999},
pages = {45-58},
zbl = {0927.05025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1084}
}
Gary Chartrand; Ping Zhang. The forcing geodetic number of a graph. Discussiones Mathematicae Graph Theory, Tome 19 (1999) pp. 45-58. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1084/
[000] [1] G. Chartrand, F. Harary and P. Zhang, The geodetic number of a graph, Networks (to appear). | Zbl 0987.05047
[001] [2] G. Chartrand, F. Harary, and P. Zhang, On the hull number of a graph, Ars Combin. (to appear). | Zbl 1064.05049