Almost-peripheral graphs

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. Almostperipheral (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 ≥ 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 Φr(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 Φ2(G) ≤ 5 holds for any non-trivial graphs and that equality holds if and only if G is a complete graph.