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The Lower and Upper Geodetic Numbers of Graphs
Author(s): 
Pages: 33-40
Year: Issue:  3
Journal: OPERATIONS RESEARCH TRANSACTIONS

Keyword:  Operation researchconvex setcomplete r-partite graphgeodesicgeodetic number;
Abstract: For any two vertices u and v in a graph G (digraph D, respectively), a u- v geodesic is a shortest path between u and v (from u to v, respectively). 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) (g(D), respectively) of a graph G (digraph D, respectively) is the minimum cardinality of a set S with I(S) = V(G)(I(S) = V(D), respectively). The lower geodetic number of G is g-(G) = min{g(D): D is an orientation of G}. The upper geodetic number of G is g+(G) = max{g(D): D is an orientation of G}. For two graphs G and H with u ∈ V(G) and v ∈ V(H), Gu + Hv is a graph obtained from G and H by adding an edge uv. The main purpose of this paper is to study the lower and upper geodetic numbers of the graph Gu + Hv.
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