In this paper, we study a Steiner tree related problem called “Terminal Steiner Tree with Bounded Edge Length”: given a set of terminal points P in a plane, one is asked to find a Steiner tree T such that any point in P is a leaf in T and the length of each edge in T is no more than a constant b. The objective of the problem is to minimize the number of Steiner points in T . This problem is mot...

The structure of Steiner quadruple system S(v, 4, 3) of full 2-rank v− 1 is considered. It is shown that there are two types (induced and singular) of such systems. It is shown that induced Steiner systems can be obtained from Steiner systems S(v, 4, 3) of 2-rank v − 2 by switching construction which is introduced here. Moreover, all non-isomorphic induced Steiner systems S(16, 4, 3) of full 2r...

Let q be a prime power and a be a positive integer such that a 2. Assume that there is a Steiner 3-(a+1, q+1, 1) design. For every v satisfying certain arithmetic conditions we can construct a Steiner 3-(va+1, q+1, 1) design for every d sufficiently large. In the case of block size 6, when q=5, this theorem yields new infinite families of Steiner 3-designs: if v is a given positive integer sati...

A minimum Steiner tree for a given set X of points is a network interconnecting the points of X having minimum possible total length. In this note we investigate various properties of minimum Steiner trees in normed planes, i.e., where the "unit disk" is an arbitrary compact convex centrally symmetric domain D having nonempty interior. We show that if the boundary of D is strictly convex and di...

A gradient-constrained minimum network T is a minimum length network, spanning a given set of nodes N in space with edges whose gradients are all no more than an upper bound m. The nodes in T but not in N are referred to as Steiner points. Such networks occur in the underground mining industry where the typical maximal gradient is about 1:7 (≈ 0.14). Because of the gradient constraint the lengt...

We study the problem of converting triangulated domains to quadrangulations, under a variety of constraints. We obtain a variety of characterizations for when a triangulation (of some structure such as a polygon, set of points, line segments or planar subdivision) admits a quadrangulation without the use of Steiner points, or with a bounded number of Steiner points. We also investigate the effe...

The Steiner tree problem is deened as follows-given a graph G = (V; E) and a subset X V of terminals, compute a minimum cost tree that includes all nodes in X. Furthermore, it is reasonable to assume that the edge costs form a metric. This problem is NP-hard and has been the study of many heuristics and algorithms. We study a generalization of this problem, where there is a \switch" cost in add...

We describe a technique to improve Smith’s branch-and-bound algorithm for the Euclidean Steiner tree problem inRd . The algorithm relies on the enumeration and optimization of full Steiner topologies for corresponding subsets of regular points. We handle the case of two Steiner points colliding during the optimization process—that is when there is a small enough distance between them—to dynamic...