Simplified constructions of almost peripheral graphs and improved embeddings into them

Sandi Klavžar, Kishori P Narayankar, Srishti P Narayankar, D Shubhalakshmi

The center and the periphery of a graph are the sets of vertices with minimum and maximum eccentricity, respectively. A graph is called almost peripheral (AP) if all its vertices but one lie in the periphery. The rAP index AP r (G) of a graph G is the smallest number of vertices needed to add to G to obtain an rAP graph in which G lies as an induced subgraph. In this paper new, simplified constructions of AP graphs are presented. It is proved that if r ≥ 2 and n ≥ 2, then AP r (K n) ≤ 4r − 3. Moreover, if G is not complete and has at least three vertices, then AP r (G) ≤ 4r − 4. In this way the previously best know bound AP r (G) ≤ 4r − 2 is improved.