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...

The novel octilinear routing paradigm (X-architecture) in VLSI design requires new approaches for the construction of Steiner trees. In this paper, we consider two versions of the shortest octilinear Steiner tree problem for a given point set K of terminals in the plane: (1) a version in the presence of hard octilinear obstacles, and (2) a version with rectangular soft obstacles. The interior o...

We extend some aspects of the theory of fractal strings and their complex dimensions from the real line to general Euclidean spaces. This is accomplished by using the explicit formulas and techniques of [La-vF4]. We use the self-similar tilings constructed in [Pe1] to define a zeta function which encodes the scaling properties of the tiling. This allows us to define a generating function for th...

This paper develops techniques for computing the minimum weight Steiner triangulation of a planar point set. We call a Steiner point P a Steiner reducing point of a planar point set X if the weight (sum of edge lengths) of a minimum weight triangulation of X ∪{P} is less than that of X. We define the Steiner reducing set St(X) to be the collection of all Steiner reducing points of X. We provide...

The main goal of this article is to present several connections between perfect codes in the Johnson scheme and designs, and provide new tools for proving Delsarte conjecture that there are no nontrivial perfect codes in the Johnson scheme. Three topics will be considered. The first is the configuration distribution which is akin to the weight distribution in the Hamming scheme. We prove that i...