In 1994, Dobrynin and Kochetova introduced the concept of degree distance DD(Γ) a connected graph Γ. Let dΓ(S) be Steiner k-distance S⊆V(Γ). The Wiener k-index or k-center indexSWk(Γ) Γ is defined by SWk(Γ)=∑|S|=kS⊆V(Γ)dΓ(S). distanceSDDk(Γ) SDDk(Γ)=∑|S|=kS⊆V(Γ)∑v∈SdegΓ(v)dΓ(S), where degΓ(v) vertex v in this paper, we consider Nordhaus–Gaddum-type results for SWk(Γ) SDDk(Γ). Upper bounds on SW...

Generalized Steiner systems GSd…t; k; v; g† were ®rst introduced by Etzion and used to construct optimal constant-weight codes over an alphabet of size g‡ 1 with minimum Hamming distance d, in which each codeword has length v and weight k. Much work has been done for the existence of generalized Steiner triple systems GS…2; 3; v; g†. However, for block size four there is not much known on GSd…2...

The power of Steiner points was studied in a number of different settings in the context of metric embeddings. Perhaps most notably in the context of probabilistic tree embeddings Bartal and Fakcharoenphol et al. [8, 9, 21] used Steiner points to devise near-optimal constructions of such embeddings. However, Konjevod et al. [24] and Gupta [22] demonstrated that Steiner points do not help in thi...

Cavernous segment internal carotid artery (CSICA) injury during endoscopic transsphenoidal surgery for pituitary tumor is rare but fatal. The aim of this study is to investigate anatomical relationship between pituitary macroadenoma and corresponding CSICA using quantitative means with a sense to improve safety of surgery.In this retrospective study, a total of 98 patients with nonfunctioning p...

We present a general rectilinear Steiner tree problem in the plane and prove that it is solvable on the Hanan grid of the input points. This result is then used to show that several variants of the ordinary rectilinear Steiner tree problem are solvable on the Hanan grid, including | but not limited to | Steiner trees for rectilinear (or iso-thetic) polygons, obstacle-avoiding Steiner trees, gro...

An algorithm for generating parity-check matrices of high-rate lowdensity parity-check codes based on permutation matrices and Steiner system S(v, 4, 2) is proposed. The estimations of rate, minimum distance and girth for derived code constructions are presented. The results of simulation of obtained code constructions for an iterative ”belief propagation” (Sum-Product) decoding algorithm, appl...

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 ...