Given a complete graph G = (V,E), where each vertex is labeled either terminal or Steiner, a distance function d : E → R, and a positive integer k, we study the problem of finding a Steiner tree T spanning all terminals and at most k Steiner vertices, such that the length of the longest edge is minimized. We first show that this problem is NP-hard and cannot be approximated within a factor 2− ε...

We extend the primal-dual approximation technique of Goemans and Williamson to the Steiner connectivity problem, a kind of Steiner tree problem in hypergraphs. This yields a (k+1)-approximation algorithm for the case that k is the minimum of the maximal number of nodes in a hyperedge minus 1 and the maximal number of terminal nodes in a hyperedge. These results require the proof of a degree pro...

Latency reduction in distributed interactive applications has been studied intensively. Such applications may have stringent latency requirements and dynamic user groups. We focus on application-layer multicast with a centralized approach to the group management. The groups are organized in overlay networks that are created using graph algorithms that create a tree structure for the group. A tr...

Given a properly face two-coloured triangulation of the graph Kn in a surface, a Steiner triple system can be constructed from each of the colour classes. The two Steiner triple systems obtained in this manner are said to form a biembedding. If the systems are isomorphic to each other it is a self-embedding. In the following, for each k ≥ 2, we construct a self-embedding of the doubled affine S...

We introduce the concept of a clique bitrade, which generalizes several known types of bitrades, including latin bitrades, Steiner T(k−1, k, v) bitrades, extended 1perfect bitrades. For a distance-regular graph, we show a one-to-one correspondence between the clique bitrades that meet the weight-distribution lower bound on the cardinality and the bipartite isometric subgraphs that are distance-...

For each positive integer n, we construct a Steiner triple system of order v = 2(3n) + 1 with no almost parallel class; that is, with no set of v−1 3 disjoint triples. In fact, we construct families of (v, k, λ)-designs with an analogous property. The only previously known examples of Steiner triple systems of order congruent to 1 (mod 6) without almost parallel classes were the projective trip...

In the Covering Steiner problem, we are given an undirected graph with edgecosts, and some subsets of vertices called groups, with each group being equipped with a non-negative integer value (called its requirement); the problem is to find a minimum-cost tree which spans at least the required number of vertices from every group. The Covering Steiner problem is a common generalization of the k-M...

In the buy-at-bulk k-Steiner tree (or rent-or-buy k-Steiner tree) problem we are given a graph G(V,E) with a set of terminals T ⊆ V including a particular vertex s called the root, and an integer k ≤ |T |. There are two cost functions on the edges of G, a buy cost b : E −→ R+ and a rent cost r : E −→ R+. The goal is to find a subtree H of G rooted at s with at least k terminals so that the cost ∑