Abstract
The center $C(G)$ and the periphery $P(G)$ of a connected graph $G$ consist of the vertices of minimum and maximum eccentricity, respectively. Almost-peripheral (AP) graphs are introduced as graphs $G$ with $|P(G)| = |V(G)| - 1$ (and $|C(G)| = 1$). AP graph of radius $r$ is called an $r$-AP graph. Several constructions of AP graph are given, in particular implying that for any $r\ge 1$, any graph can be embedded as an induced subgraph into some $r$-AP graph. A decomposition of AP-graphs that contain cut-vertices is presented. The $r$-embedding index $\Phi_{r}(G)$ of a graph $G$ is introduced as the minimum number of vertices which have to be added to $G$ such that the obtained graph is an $r$-AP graph. It is proved that $\Phi_{2}(G)\le 5$ holds for any non-trivial graphs and that equality holds if and only if $G$ is a complete graph.
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