Peripheral and eccentric vertices in graphs (Q1205346)

Let \(r\) and \(d\) denote the radius and diameter of a graph \(G\) and \(\text{P}(G)\) denote the set of peripheral vertices of \(G\). A vertex \(v\) such that \(d(v,c)=r\) for some central vertex \(c\) of \(G\), is defined, by the authors, to be an eccentric vertex. Let \(\text{EC}(G)\) denote the set of eccentric vertices of \(G\). Using some interesting constructions the authors prove, in the last section, that all types of set inclusion relations are possible between \(\text{P}(G)\) and \(\text{EC}(G)\). In section 1 they prove that \(\text{EC}(G)=\text{P}(G)\) when either (i) \(G\) is a simple connected graph with a unique central vertex \(v\), such that every block containing \(v\) is complete or (ii) \(G\) is a non-selfcentered graph with \(\text{C}(G)\) contained in a complete block, and \(|\text{C}(G)|\geq 2\). In the middle section they obtain necessary and sufficient conditions for \(\text{EC}(G)=\text{P}(G)\) when \(d=2r\) or \(d=2r-1\).