In the construction of a communication network, the (Euclidean) length of the network is an important but not unique factor determining the cost of the network. Among many possible network models, Gilbert proposed a flow-dependent model in which flow demands are assigned between each pair of points in a given point set A, and the cost per unit length of a link in the network is a function of th...

In this paper, Steiner and non-Steiner 2-(v, 3) trades of minimum volume are considered. It is shown that these trades are composed of a union of some Pasch configurations and possibly some 2-(v, 3) trades with 6 ≤ v ≤ 10. We determine the number of non-isomorphic Steiner 2-(v, 3) trades of minimum volume. As for non-Steiner trades the same thing is done for all vs, except for v ≡ 5 (mod 6).

Given a convex polygon P in the plane and a positive integer n, we consider the problem of generating a length-uniform triangular mesh for the interior of P using n Steiner points. More specifically, we want to find both a set Sn of n points inside P , and a triangulation of P using Sn, with respect to the following minimization criteria: (1) ratio of the maximum edge length to the minimum one,...

The classification, up to isomorphism, of all self-embedding monomial power permutations of Hamming Steiner triple systems of order n = 2m − 1 for small m, m ≤ 22, is given. As far as we know, for m ∈ {5, 7, 11, 13, 17, 19}, all given self-embeddings in closed surfaces are new. Moreover, they are cyclic for all m and nonorientable at least for all m ≤ 19. For any non prime m, the nonexistence o...

We generalized the Steiner’s shortest line problem and found its connection with the concepts in classical field theory. We solved the generalized Steiner’s problem by introducing a conservative potential and a dissipative force in the field and gave a computing method by using a testing point and a corresponding iterative curve. 1 Steiner’s Problem In the R Euclidean space, there are n points ...

A Steiner quadruple system (X, B) is said to be (1, 2)-resolvable if its blocks can be partitioned into r parts such that each point of X occurs in exactly two blocks in each part. The necessary condition for the existence of (1, 2)-resolvable Steiner quadruple systems RSQS(1, 2, v)s is v ≡ 2 or 10 (mod 12). Hartman and Phelps in [A. Hartman, K.T. Phelps, Steiner quadruple systems, in: J.H. Din...

A full Steiner tree T for a given set of points P is defined to be linear if all Steiner points lie on one path called the trunk of T . A (nonfull) Steiner tree is linear if it is a degeneracy of a full linear Steiner tree. Suppose P is a simple polygonal line. Roughly speaking, T is similar to P if its trunk turns to the left or right when P does. P is a left-turn (or right-turn) polygonal spi...

Recently a new graph convexity was introduced, arising from Steiner intervals in graphs that are a natural generalization of geodesic intervals. The Steiner tree of a set W on k vertices in a connected graph G is a tree with the smallest number of edges in G that contains all vertices of W . The Steiner interval I(W ) of W consists of all vertices in G that lie on some Steiner tree with respect...

For any 1-error-correcting binary code C of length m we will construct a 1-perfect binary code P (C) of length n = 2 − 1 such that fixing the last n − m coordinates by zeroes in P (C) gives C. In particular, any complete or partial Steiner triple system (or any other system that forms a 1-code) can always be embedded in a 1-perfect code of some length (compare with [13]). Since the weight-3 wor...

In this paper, we consider a variant of the well-known Steiner tree problem. Given a complete graph G = (V, E) with a cost function c : E → R and two subsets R and R satisfying R ⊂ R ⊆ V , a selected-internal Steiner tree is a Steiner tree which contains (or spans) all the vertices in R such that each vertex in R cannot be a leaf. The selected-internal Steiner tree problem is to find a selected...