Steiner centers and Steiner medians of graphs

HT-graphs: centers, connected r-domination and Steiner trees

HT-graphs have been introduced in [11] and investigated with respect to location problems on graphs. In this paper two new characterizations of these graphs are given and then it is shown that the central vertex, connected r-domination and Steiner trees problems are linear or almost linear time solvable in HT-graphs.

Steiner distance stable graphs

Let G be a connected graph and S a nonempty set of vertices of G. Then the Steiner distance d,(S) of S is the smallest number of edges in a connected subgraph of G that contains S. Let k, I, s and m be nonnegative integers with m > s > 2 and k and I not both 0. Then a connected graph G is said to be k-vertex I-edge (s,m)-Steiner distance stable, if for every set S of s vertices of G with d,(S) ...

Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I (S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S = {u, v}, then I (S) = I [u, v] is called the interval between u and v and consists of all vertices that lie on some shortest u–v path in G. T...

Centers and medians of distance-hereditary graphs

A graph is distance-hereditary if the distance between any two vertices in a connected induced subgraph is the same as in the original graph. In this paper, we study metric properties of distance-hereditary graphs. In particular, we determine the structures of centers and medians of distance-hereditary and related graphs. The relations between eccentricity, radius, and diameter of such graphs a...

Steiner Trees and Steiner Forests