The Steiner diameter of a graph

The Steiner distance of a graphý, ýintroduced by Chartrandý, ýOellermanný, ýTian and Zou in 1989ý, ýis a natural generalization of theý ýconcept of classical graph distanceý. ýFor a connected graph G ofý ýorder at least 2 and S⊆V(G)ý, ýthe Steinerý ýdistance d(S) among the vertices of S is the minimum size amongý ýall connected subgraphs whose vertex sets contain Sý. ýLet n,k beý ýtwo integers with 2≤k≤ný. ýThen the Steinerý ýk-eccentricity ek(v) of a vertex v of G is defined byý ýek(v)=max{d(S)|S⊆V(G)ý,|S|=ký,andv∈Sýý}ý. ýFurthermoreý, ýthe Steiner k-diameter of G isý ýsdiamk(G)=max{ek(v)|ýv∈V(G)}ý. ýIn 2011ý, ýChartrandý, ýOkamoto and Zhang showed that k−1≤sdiamk(G)≤n−1ý. ýIn thisý ýpaperý, ýgraphs with sdiam3(G)=2,3,n−1 are characterizedý, ýrespectivelyý. ýWe also consider the Nordhaus-Gaddum-type results forý ýthe parameter sdiamk(G)ý. ýWe determine sharp upper and lowerý ýbounds of sdiamk(G)뇘骚(G¯¯¯¯) and sdiamk(G)⋅ýýsdiamk(G¯¯¯¯) for a graph G of order ný. ýSomeý ýgraph classes attaining these bounds are also given.