In the Steiner Networks Activation problem we are given a graph G = (V,E), S ⊆ V , a family {fuv(xu, xv) : uv ∈ E} of monotone non-decreasing activating functions from R2+ to {0, 1} each, and connectivity requirements {r(u, v) : u, v ∈ V }. The goal is to find a weight assignment w = {wv : v ∈ V } of minimum total weight w(V ) = ∑ v∈V wv, such that: for all u, v ∈ V , the activated graph Gw = (...

This paper defines a class of designs which generalise t-designs, resolvable designs, and orthogonal arrays. For the parameters t = 2, k = 3 and λ = 1, the designs in the class consist of Steiner triple systems, Latin squares, and 1-factorisations of complete graphs. For other values of t and k, we obtain t-designs, Kirkman systems, large sets of Steiner triple systems, sets of mutually orthogo...

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 ...

We consider bipartite graphs on m red points and n blue points, where m ^ n, and prove that, for any fixed k, almost all such graphs (labelled or unlabelled) are k-connected as n -» oo, provided m > C logn, where C depends on k. If Tmn is the number of such unlabelled graphs, we show that Tmn ~ 2 /(m \n!). If T'mn is the number of such unlabelled graphs with the colours removed, then T'mn ~ Tmn...

Let $G$ be a connected graph, and let $D[G]$ denote the double graph of $G$. In this paper, we first derive closed-form formulas for different distance based topological indices for $D[G]$ in terms of that of $G$. Finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topologica...

In [4] we introduced a product construction for cyclic Steiner quadruple systems. The purpose of this paper is to modify the construction so that it is applicable to cyclic Steiner 2-designs. A cyclic Steiner 2-design is a Steiner system s(t, k, u), 2 d t < k < u, in which t = 2 and which has an automorphism of order u. We denote such a system by CS(2, k, u). Under suitable conditions we show h...

We consider the Steiner Multicut problem, which asks, given an undirected graph G, a collection T = {T1, . . . , Tt}, Ti ⊆ V (G), of terminal sets of size at most p, and an integer k, whether there is a set S of at most k edges or nodes such that of each set Ti at least one pair of terminals is in different connected components of G \S. This problem generalizes several well-studied graph cut pr...

The Steiner tree problem which is known to be NP-Complete is the following. Given a weighted undirected graph G = (V;E), and a set X V of terminals, the objective is to nd a tree of minimum cost which connects all the terminals. If the graph is directed, in addition to X, we are given a root r 2 V , and the objective is to nd a minimum cost arborescence which connects the root to each of the te...

a topological index of a molecular graph g is a numeric quantity related to g which isinvariant under symmetry properties of g. in this paper we obtain the randić, geometricarithmetic,first and second zagreb indices , first and second zagreb coindices of tensorproduct of two graphs and then the harary, schultz and modified schultz indices of tensorproduct of a graph g with complete graph of ord...

We present new functional equations for the species of plane and of planar (in the sense of Harary and Palmer, 1973) 2-trees and some associated pointed species. We then deduce the explicit molecular expansion of these species, i.e. a classification of their structures according to their stabilizers. There result explicit formulas in terms of Catalan numbers for their associated generating seri...