Given two sets of points in the plane, P of n terminals and S of m Steiner points, a Steiner tree of P is a tree spanning all points of P and some (or none or all) points of S. A Steiner tree with length of longest edge minimized is called a bottleneck Steiner tree. In this paper, we study the Euclidean bottleneck Steiner tree problem: given two sets, P and S, and a positive integer k ≤ m, find...

Steiner systems S(2, k, v) For k ≥ 3, a Steiner system S(2, k, v) is usually defined as a pair (V,B), where V is a set of cardinality v of points and B is a set of k-element subsets of V , usually called blocks, or lines if the system has some geometric significance, with the property that each pair of points is contained in precisely one block. For example, to construct a Steiner system S(2, 3...

We study the Steiner k-eccentricity on trees, which generalizes previous one in paper [On average 3-eccentricity of arXiv:2005.10319]. achieve much stronger properties for k-ecc tree than that paper. Based this, a linear time algorithm is devised to calculate vertex tree. On other hand, lower and upper bounds index order n are established based novel technique quite different also easier follow...

A Steiner System, denoted S(t, k, v), is a vertex set X containing v vertices, and a collection of subsets of X of size k, called blocks, such that every t vertices from X are in exactly one of the blocks. A Steiner Triple System, or STS, is a special case of a Steiner System where t = 2, k = 3 and v = 1 or 3 (mod6) [7]. A Bi-Steiner Triple System, or BSTS, is a Steiner Triple System with the v...

Ž . A Steiner minimum tree SMT in the rectilinear plane is the shortest length tree interconnecting a set of points, called the regular points, possibly using Ž . additional vertices. A k-size Steiner minimum tree kSMT is one that can be split into components where all regular points are leaves and all components have at most k leaves. The k-Steiner ratio in the rectilinear plane, r , is the in...

We present the first exact algorithm for constructing minimum bottleneck 2-connected Steiner networks containing at most k Steiner points, where k > 2 is a constant integer. Given a set of n terminals embedded in the Euclidean plane, the objective of the problem is to find the locations of the Steiner points, and the topology of a 2-connected graph Nk spanning the Steiner points and the termina...

Given a graph G = (V; E) and a positive integer k, the Phylogenetic k-Root Problem asks for a (unrooted) tree T without degree-2 nodes such that its leaves are labeled by V and (u; v) 2 E if and only if dT (u; v) k. If the vertices in V are also allowed to be internal nodes in T , then we have the Steiner k-Root Problem. Moreover, if a particular subset S of V are required to be internal nodes ...

A partial Steiner (k, l)-system is a k-uniform hypergraph G with the property that every l-element subset of V is contained in at most one edge of G . In this paper we show that for given k, l and t there exists a partial Steiner (k, l)-system such that whenever an l-element subset from every edge is chosen, the resulting l-uniform hypergraph contains a clique of size t. As the main result of t...