Steiner distance in graphs

Abstract

For a nonempty set S of vertices of a connected graph G, the distance d(S) of S is the minimum size of a connected subgraph whose vertex set contains S. For integers n and p with 2 5^ n ^. p, the minimum size of a graph G of order p is determined for which d(S) = n — 1 for all sets S of vertices of G having |S| = n. For a connected graph G of order/? and integer n with 2 S^n^p, the n-eccentricity of a vertex v of G is the maximum value of d(S) over all «->._ V(G) with v in S and |S| -= n. The minimum n-eccentricity radw G is called the w-radius of G and the maximum n-eccentricity diam., G is its n-diameter. It is shown that diamn T 5* 5j [n/(w — 1)] rad,, T for every tree T of order p with 2^«^/>. For a graph G of order p the sequence diam2 G, diam3 G,..., diamp G is called the diameter sequence of G. In the case of trees, the n-radius and n-diameter are investigated and the diameter sequences of trees are characterized. 1.