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. The objective of the problem is – given a set of n terminals embedded in the Euclidean plane – to find the locations of the Steiner points, and the topology of a 2-connected graph Nk spanning the Steiner points and the term...

‎The Steiner distance of a graph‎, ‎introduced by Chartrand‎, ‎Oellermann‎, ‎Tian and Zou in 1989‎, ‎is a natural generalization of the‎ ‎concept of classical graph distance‎. ‎For a connected graph $G$ of‎ ‎order at least $2$ and $Ssubseteq V(G)$‎, ‎the Steiner‎ ‎distance $d(S)$ among the vertices of $S$ is the minimum size among‎ ‎all connected subgraphs whose vertex sets contain $S$‎. ‎Let $...

We introduce a temporal Steiner network problem in which a graph, as well as changes to its edges and/or vertices over a set of discrete times, are given as input; the goal is to find a minimal subgraph satisfying a set of k time-sensitive connectivity demands. We show that this problem, k-Temporal Steiner Network (k-TSN), is NP-hard to approximate to a factor of k − , for every fixed k ≥ 2 and...

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...

A {it topological index} of a graph is a real number related to the graph; it does not depend on labeling or pictorial representation of a graph. In this paper, we present the upper bounds for the product version of reciprocal degree distance of the tensor product, join and strong product of two graphs in terms of other graph invariants including the Harary index and Zagreb indices.

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 ...