on the Steiner Median of a Tree

Abstract Let S be a nonempty set of vertices of a connected graph G. Then the Steiner distance of S is the minimum size (number of edges) of a connected subgraph of G containing S. Let n ⩾ 2 be an integer and suppose that G has at least n vertices. Then the Steiner n-distance of a vertex v of G is defined to be the sum of the Steiner distances of all sets of n vertices that include v. The Steiner n-median Mn(G) of G is the subgraph induced by the vertices of minimum Steiner n-distance. We show that the Steiner n-median of a tree is connected and determine those trees that are Steiner n-medians of trees, and show that if T is a tree with more than n vertices, then Mn(T) ⊂- Mn + 1(T). Further, a O (¦V(T)¦) algorithm for finding the Steiner n-median of a tree T is presented and a O (n¦V(T)¦ 2 ) algorithm for finding the Steiner n-distances of all vertices in tree T is described. For a connected graph G of order p ⩾ n, the n-median value of G is the least Steiner n-distance of any of its vertices. For p ⩾ 2n − 1, sharp upper and lower bounds for the n-median values of trees of order p are given, and it is shown that among all trees of a given order, the path has maximum n-median value.