For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W -tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steine...

The Clustered Steiner tree problem is a variant of Steiner minimum tree problem. The required vertices are partitioned into clusters, and the subtrees spanning different clusters must be disjoint in a feasible clustered tree. In this paper we show that the Steiner ratio of the cluster Steiner tree problem is three, where the Steiner ratio is defined as the largest possible ratio of the minimal ...

Given a graph G = (V, E) with a length function on edges and a subset R of V , the full Steiner tree is defined to be a Steiner tree in G with all the vertices of R as its leaves. Then the full Steiner tree problem is to find a full Steiner tree in G with minimum length, and the bottleneck full Steiner tree problem is to find a full Steiner tree T in G such that the length of the largest edge i...

The Steiner connectivity problem is a generalization of the Steiner tree problem. It consists in finding a minimum cost set of simple paths to connect a subset of nodes in an undirected graph. We show that polyhedral and algorithmic results on the Steiner tree problem carry over to the Steiner connectivity problem; namely, the Steiner cut and the Steiner partition inequalities, as well as the a...

An infinite countable Steiner triple system is called universal if any countable Steiner triple system can be embedded into it. The main result of this paper is the proof of non-existence of a universal Steiner triple system. The fact is proven by constructing a family S of size 2 of infinite countable Steiner triple systems so that no finite Steiner triple system can be embedded into any of th...

We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (St...

For a connected graph G = (V,E), a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W -tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. For a minimum Steiner set W of G, a subset T ⊆ W is called a forcing subset for W if W is the unique minimum Steine...

Steiner systems are a fascinating topic of combinatorics. The most studied Steiner systems are S(2,3,v) (Steiner triple systems), S(3,4,v) (Steiner quadruple systems), and S(2,4,v). There are a few infinite families of Steiner systems S(2,4,v) in the literature. The objective of this paper is to present an infinite family of Steiner systems S(2,4,2m) for all m ≡ 2 (mod 4)≥ 6 from cyclic codes. ...

Given an edge-weighted graph G = (V,E) and a subset R of V , a Steiner tree of G is a tree which spans all the vertices in R. A full Steiner tree is a Steiner tree which has all the vertices of R as its leaves. The full Steiner tree problem is to find a full Steiner tree of G with minimum weight. In this paper we present a 20-approximation algorithm for the full Steiner tree problem when G is a...

We consider the problem of constructing Steiner minimum trees for a metric defined by a polygonal unit circle (corresponding to σ ≥ 2 weighted legal orientations in the plane). A linear-time algorithm to enumerate all angle configurations for degree three Steiner points is given. We provide a simple proof that the angle configuration for a Steiner point extends to all Steiner points in a full S...