Abstract
For any graphs Gi and G2, and an integer d (2 ≤ d ≤ r(G2)), define β(G1,G2) (or β(G1, G2; d)) to be the minimum number of vertices of the graph H which contains G1 as its center and G2 as its periphery (and dia(H) = d, respectively). In this paper, the values of β(G1,G2) and the upper bounds for β(G1, G2; d) are obtained when G2 is not 3-self-centered.
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