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Discussiones Mathematicae Graph Theory 20(1) (2000) 129-138
DOI: https://doi.org/10.7151/dmgt.1112

GEODETIC SETS IN GRAPHS

Abstract

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.

Keywords: geodetic set, geodetic number, upper geodetic number.

AMS Subject Classification: 05C12.

References

[1]	G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks (to appear).
[2]	G. Chartrand and L. Lesniak, Graphs & Digraphs (third edition, Chapman & Hall, New York, 1996).
[3]	G. Chartrand and P. Zhang, The forcing geodetic number of a graph, Discuss. Math. Graph Theory 19 (1999) 45-58, doi: 10.7151/dmgt.1084.
[4]	G. Chartrand and P. Zhang, The geodetic number of an oriented graph, European J. Combin. 21 (2000) 181-189, doi: 10.1006/eujc.1999.0301.
[5]	F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969).
[6]	H.M. Mulder, The Interval Function of a Graph (Mathematisch Centrum, Amsterdam, 1980).
[7]	L. Nebeský, A characterization of the interval function of a connected graph, Czech. Math. J. 44 (119) (1994) 173-178.
[8]	L. Nebeský, Characterizing of the interval function of a connected graph, Math. Bohem. 123 (1998) 137-144.

Received 27 September 1999
Revised 26 January 2000