The minimal closed geodetic numbers of graphs - Aniversario, Imelda S.,; Jamil, Ferdinand P., (Utilitas Mathematica, 2009)
Given two vertices u and υ of a connected graph G., the closed interval J G[u, υ] is the set of all vertices lying in some u-υ geodesic in Q. If S ⊆ V(G), then I G[S] = ∪{I G[u,υ]: u,υ ∈ S}. A set S of vertices in G is called a geodetic cover of G if I G[S] - V(G). The geodetic number gn(G) of G is the minimum cardinality of a geodetic cover of G.A. geodetic cover of smallest cardinality is called a geodetic basis of G. Suppose that in constructing a geodetic cover of G., we select a vertex υ 1 and let S 1 = {υ 1}. Select a vertex υ 2 ≠ υ 1 and let S 2 = {υ 1, υ 2}. Then successively select vertex υ 1 ∉ I G[S i-1] and let S i = {υ 1, υ 2,υ i}. The closed geodetic number cgn(G) and the upper closed geodetic number ucgn(G) of G is the smallest and the largest k., respectively, for which selection of υ k in the given manner makes I G[S k] = V(G). A closed geodetic cover S of G is a minimal closed geodetic cover of G if no proper subset of S is a closed geodetic cover of G. The minimal closed geodetic number mcgn(G) is the maximum cardinality of a minimal closed geodetic cover of G. In this paper, it is shown that ucgn(G) = mcgn(G) if and only if G is complete, while cgn{G) and mcgn(G) coincide among extreme geodesic graphs G. Moreover, for complete bipartite graphs K m,n cgn(K m,n) = mcgn(K m,n) if and only if m = n. More interesting, for every triple a, 6, c ∈ Z +, with 2 ≤ a < b < c, a, b, and c are realizable as closed geodetic number, minimal closed geodetic number, and upper closed geodetic number, respectively, of a connected graph. We also determine here the minimal closed geodetic numbers of graphs resulting from the join of graphs.