The problem of determining the largest number f(n; κ ≤ ) of edges for graphs with n vertices and maximal local connectivity at most was considered by Bollobás. Li et al. studied the largest number f(n; κ3 ≤ 2) of edges for graphs with n vertices and at most two internally disjoint Steiner trees connecting any three vertices. In this paper, we further study the largest number f(n; κk = 1) of edg...

In the Fixed Cost k-Flow problem, we are given a graph G = (V,E) with edge-capacities {ue | e ∈ E} and edge-costs {ce | e ∈ E}, source-sink pair s, t ∈ V , and an integer k. The goal is to find a minimum cost subgraph H of G such that the minimum capacity of an st-cut in H is at least k. We show that the Group Steiner on Trees problem is a special case of Fixed Cost k-Flow. This implies the fir...

We survey approximation algorithms and hardness results for versions of the Generalized Steiner Network (GSN) problem in which we seek to find a low cost subgraph (where the cost of a subgraph is the sum of the costs of its edges) that satisfies prescribed connectivity requirements. These problems include the following well known problems: min-cost k-flow, min-cost spanning tree, traveling sale...

Directed Steiner problems are fundamental problems in Combinatorial Optimization and Theoretical Computer Science. An important problem in this genre is the k-edge connected directed Steiner tree (k-DST) problem. In this problem, we are given a directed graph G on n vertices with edge-costs, a root vertex r, a set of h terminals T and an integer k. The goal is to find a min-cost subgraph H ⊆ G ...

C-loops are loops satisfying the identity x(y · yz) = (xy · y)z. We develop the theory of extensions of C-loops, and characterize all nuclear extensions provided the nucleus is an abelian group. C-loops with central squares have very transparent extensions; they can be built from small blocks arising from the underlying Steiner triple system. Using these extensions, we decide for which abelian ...

We study the Online Steiner Tree Leasing (OSTL) problem, defined in a weighted undirected graph with a distinguished root node r. There is a known set L of available lease types, where each type ` ∈ L is characterized by its duration D` and cost factor C`. As an input, an online algorithm is given a sequence of terminals and has to connect them to the root r using leased edges. An edge of lengt...

Given a connected graph G and a terminal set R ⊆ V (G), Steiner tree asks for a tree that includes all of R with at most r edges for some integer r ≥ 0. It is known from [ND12,Garey et. al [1]] that Steiner tree is NP-complete in general graphs. Split graph is a graph which can be partitioned into a clique and an independent set. K. White et. al [2] has established that Steiner tree in split gr...

Given a directed graph G and a list (s1, t1), . . . , (sk, tk) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed si → ti path for every 1 ≤ i ≤ k. The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t1, . . . , tk) is known to be fixed-parameter tractable parameterized by the number of term...

Results of Frankl and RR odl and of Pippenger and Spencer show that uniform hypergraphs which are almost regular and have small maximal pair degrees (codegrees) contain collections of pairwise disjoint edges (packings) which cover all but o(n) of the n vertices. Here we show, in particular, that regular uniform hypergraphs for which the ratio of degree to maximum codegree is n " , for some " > ...

Given an undirected graph with nonnegative edge-costs, a subset of nodes of size k called the terminals, and an integer q between 1 and k, the minimum q-Steiner forest problem is to find a forest of minimum cost with at most q trees that spans all the terminals. When q = 1, we have the classical minimum-cost Steiner tree problem on networks. We adapt a primal-dual approximation algorithm for th...