The geodesic and geodesic interval, namely the set of all vertices lying on geodesics between a pair of vertices in a connected graph, is a part of folklore in metric graph theory. It is also known that Steiner tree of a (multi) set with k (k > 2) vertices, generalizes geodesics. In [1] the authors studied the k-Steiner intervals S(u1, u2, . . . , uk) on connected graphs (k ≥ 3) as the k-ary ge...

This paper describes a biased random-key genetic algorithm for a real-world wireless backhaul network design problem. This is a novel problem, closely related to variants of the Steiner tree problem and the facility location problem. Given a parameter h, we want to build a forest where each tree has at most h hops from the demand nodes, where traffic originates, to the root nodes where each tre...

We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G = (V, E) and an integer k, decide whether there exists F ⊆ V , |F | ≤ k, such that G[V \ F ] is a forest and G[F ] is connected. We show that Connected ...

Let G = (V; E) be a graph and T V be a node set. We call an edge set S a Steiner tree with respect to T if S connects all pairs of nodes in T. In this paper we address the following problem, which we call the weighted Steiner tree packing problem. Given a graph G = (V; E) with edge weights w e , edge capacities a Steiner tree with respect to T k , at most c e of these edge sets use edge e for e...

let g be a connected simple (molecular) graph. the distance d(u, v) between two vertices u and v of g is equal to the length of a shortest path that connects u and v. in this paper we compute some distance based topological indices of h-phenylenic nanotorus. at first we obtain an exactformula for the wiener index. as application we calculate the schultz index and modified schultz index of this ...

Given an undirected graph with weights associated with its nodes, the Steiner tree problem consists in nding a minimum weight subgraph spanning a given subset of nodes (terminals) of the original graph. In this paper, we describe an improved tabu search algorithm for the Steiner problem in graphs, based on a neighborhood deened by insertions and eliminations of Steiner nodes. Move estimations, ...

In the Covering Steiner problem, we are given an undirected graph with edgecosts, and some subsets of vertices called groups, with each group being equipped with a non-negative integer value (called its requirement); the problem is to find a minimum-cost tree which spans at least the required number of vertices from every group. The Covering Steiner problem is a common generalization of the k-M...

We consider the Shallow-Light Steiner Network problem from a fixed-parameter perspective. Given a graph G, a distance bound L, and p pairs of vertices (s1, t1), . . . , (sp, tp), the objective is to find a minimum-cost subgraph G′ such that si and ti have distance at most L in G′ (for every i ∈ [p]). Our main result is on the fixed-parameter tractability of this problem with parameter p. We exa...

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 an edge-weighted undirected graph G and a subset of “required” vertices R ⊆ V (G), called the terminals, we want to find a minor G′ with possibly different edge-weights, that retains distances between all terminal-pairs exactly, and is as small as possible. We prove that every graph G with n vertices and k terminals can be reduced (in this sense) to a minor G′ with O(k4) vertices and edge...