Abstract
For every two vertices u and v in a graph G, a u-v geodesic is a shortest path between u and v. Let I(u, v) denote the set of all vertices lying on a u-v geodesic. For a vertex subset S, let I(S) denote the union of all I(u, v) for u, v ∈ S. The geodetic number g(G) of a graph G is the minimum cardinality of a set S with I(S) = V (G). For a digraph D, there is analogous terminology for the geodetic number g(D). The geodetic spectrum of a graph G, denoted by S(G), is the set of geodetic numbers of all orientations of graph G. The lower geodetic number is g −(G) = minS(G) and the upper geodetic number is g +(G) = maxS(G). The main purpose of this paper is to study the relations among g(G), g −(G) and g +(G) for connected graphs G. In addition, a sufficient and necessary condition for the equality of g(G) and g(G × K 2) is presented, which improves a result of Chartrand, Harary and Zhang.
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This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10301010, 60673048) and the Science and Technology Commission of Shanghai Municipality (Grant No. 04JC14031)
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Lu, Ch. The geodetic numbers of graphs and digraphs. SCI CHINA SER A 50, 1163–1172 (2007). https://doi.org/10.1007/s11425-007-0048-x
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DOI: https://doi.org/10.1007/s11425-007-0048-x