In the Directed Steiner Tree problem, we are given a directed graph G = (V,E) with edge costs, a root vertex r ∈ V , and a terminal set X ⊆ V . The goal is to find the cheapest subset of edges that contains an r-t path for every terminal t ∈ X. The only known polylogarithmic approximations for Directed Steiner Tree run in quasi-polynomial time and the best polynomial time approximations only ac...

The minimum weight codewords in the Preparata code of length n = 4 m are utilized for the construction of an innnite family of Steiner S(4; f5; 6g; 4 m + 1) designs for any m 2. A t-wise balanced design with parameters t-(v; K;) is a pair (X; B) where X is a set of v points and B is a collection of subsets of X (called blocks) with sizes from the set K, such that every t-subset of X is containe...

For a graph G and a positive integer k, the k-power of G is the graph G with V (G) as its vertex set and {(u, v)|u, v ∈ V (G), dG(u, v) ≤ k} as its edge set where dG(u, v) is the distance between u and v in graph G. The k-Steiner root problem on a graph G asks for a tree T with V (G) ⊆ V (T ) and G is the subgraph of T k induced by V (G). If such a tree T exists, we call it a k-Steiner root of ...

The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paperweprovide anupper boundof theHarary index in terms of the vertex or edge connectivity of a graph. We characterize the unique graph with themaximumHarary index among all graphs with a given number of cut vertices or vertex connectivity or edge connectivity. In ad...

We study a bottleneck Steiner tree problem: given a set P = {p1,p2, . . . , pn} of n terminals in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length of the longest edges in the tree is minimized. The problem has applications in the design of wireless communication networks. We give a ratio-1.866 approximation algorithm for the pr...

This paper addresses the competitiveness of online algorithmsfor two Steiner Tree problems. In the online setting, requests for k ter-minals appear sequentially, and the algorithm must maintain a feasible,incremental solution at all times. In the first problem, the underlyinggraph is directed and has bounded asymmetry, namely the maximumweight of antiparallel links in the gr...

In this note, we prove that there does not exist a Steiner (v, k, 2) trade of volume m, where m is odd, 2k + 3 ~ m ~ 3k 4, and k ~ 7. This completes the spectrum problem for Steiner (v, k, 2) trades.

We introduce a modification of the Harary index where the contributions of vertex pairs are weighted by the sum of their degrees. After establishing basic mathematical properties of the new invariant, we proceed by finding the extremal graphs and investigating its behavior under several standard graph products.

The concept of pendant tree-connectivity was introduced by Hager in 1985. For a graph G = (V,E) and a set S ⊆ V (G) of at least two vertices, an S-Steiner tree or a Steiner tree connecting S (or simply, an S-tree) is a subgraph T = (V ′, E ′) of G that is a tree with S ⊆ V ′. For an S-Steiner tree, if the degree of each vertex in S is equal to one, then this tree is called a pendant S-Steiner t...