On the Steiner Periphery and Steiner Eccentricity of a Graph

Let G by a connected graph and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum number of edges in a connected subgraph of G that contains S. If n ≥ 2 is an integer, and G is a graph with at least n vertices, then the n-eccentricity e(n;v) of a vertex v is defined as max{d(S)|S ⊆ V(G), |S| = n and v ∈ S}. The n-radius rad _n G is the minimum n-eccentricity over all vertices of G while the n-diameter is the maximum n-eccentricity over all vertices of G. The subgraph induced by those vertices with n- eccentricity rad _n G is called the n-centre C(n;G) of G and the subgraph induced by those vertices with n- eccentricity diam _n G is called the n-periphery P(n;G) of G. A vertex v is called n-eccentric if there exists a vertex u in C(n;G) and a set S of n vertices that contains both u and v such that d(S) = e(n;u) = rad _n G. The subgraph of G induced by all n-eccentric vertices of G is called the n-eccentricity of G and is denoted by EC(n;G). It is shown that for a tree T of order at least n, P(n;T) = EC(n;T). Further, it is shown that all possible set-inclusion relations between P(n;G) and EC(n;G) may occur if G is not a tree.