Let G be a connected graph of order p and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set contains S. If n is an integer, 2 ≤ n ≤ p, the Steiner n-diameter, diamn(G), of G is the maximum Steiner distance of any n-subset of vertices of G. We give a bound on diamn(G) for a graph G in terms of the order of G a...

We prove a limit theorem for the total Steiner k-distance of a random b-ary recursive tree with weighted edges. The total Steiner k-distance is the sum of all Steiner k-distances in a tree and it generalises the Wiener index. The limit theorem is obtained by using a limit theorem in the general setting of the contraction method. In order to use the contraction method we prove a recursion formul...

The Wiener index W (G) of a connected graph G is defined as W (G) = ∑ u,v∈V (G) dG(u, v) where dG(u, v) is the distance between the vertices u and v of G. For S ⊆ V (G), the Steiner distance d(S) of the vertices of S is the minimum size of a connected subgraph of G whose vertex set is S. The k-th Steiner Wiener index SWk(G) of G is defined as SWk(G) = ∑ S⊆V (G) |S|=k d(S). We establish expressi...

The concept of pendant tree-connectivity was introduced by Hager in 1985. For a graph G = (V,E) and a set S ⊆ V (G) of at least two vertices, an S-Steiner tree or a Steiner tree connecting S (or simply, an S-tree) is a subgraph T = (V ′, E ′) of G that is a tree with S ⊆ V ′. For an S-Steiner tree, if the degree of each vertex in S is equal to one, then this tree is called a pendant S-Steiner t...

Given a point set K of terminals in the plane, the octilinear Steiner tree problem is to find a shortest tree that interconnects all terminals and edges run either in horizontal, vertical, or ±45 diagonal direction. This problem is fundamental for the novel octilinear routing paradigm in VLSI design, the socalled X-architecture. As the related rectilinear and the Euclidian Steiner tree problem ...

We introduce and study a new Steiner tree problem variation called the bursty Steiner tree problem where new nodes arrive into bursts. This is an online problem which becomes the well-known online Steiner tree problem if the number of nodes in each burst is exactly one and becomes the classical Steiner tree problem if all the nodes that need to be connected appear in a single burst. In undirect...

Let n, k, and t be integers satisfying n > k > t ≥ 2. A Steiner system with parameters t, k, and n is a k-uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for t ≥ 6. In this note we prove that for every k > t ≥ 2 and sufficiently large n,...

An Sq[t,k,v] q-Steiner system is a collection of k-dimensional subspaces of the v-dimensional vector space Fq over the finite field Fq with q elements, called blocks, such that each t-dimensional subspace of Fq is contained in exactly one block. The smallest admissible parameters for which a q-Steiner system could exist is S2[2,3,7]. Up to now the issue whether q-Steiner systems with these para...

Abstract We have studied the Steiner tree problem using six-pin soap films in detail. We extend the existing method of experimental realisation of Steiner trees in n-terminal problem through soap films to observe new non-minimal Steiner trees. We also produced spanning tree configurations for the first time by our method. Experimentally, by varying the pin diameter, we have achieved these new s...

Given a set of k-colored points in the plane, we consider the problem of finding k trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For k = 1, this is the well-known Euclidean Steiner tree problem. For general k, a kρ-approximation algorithm is known, where ρ ≤ 1.21 is the Steiner ratio. We present a PTAS ...