An inverse minimum spanning tree problem is to make the least modification on the edge weights such that a predetermined spanning tree is a minimum spanning tree with respect to the new edge weights. In this paper, the inverse minimum spanning tree problem with stochastic edge weights is investigated. The concept of α-minimum spanning tree is initiated, and subsequently an α-minimum spanning tr...

The robust spanning tree problem is a variation, motivated by telecommunications applications, of the classic minimum spanning tree problem. In the robust spanning tree problem edge costs lie in an interval instead of having a fixed value. Interval numbers model uncertainty about the exact cost values. A robust spanning tree is a spanning tree whose total cost minimizes the maximum deviation fr...

This first algorithm is quite simple. (Though this was probably known earlier, its proof can be found in Prof. Indyk’s 1999 paper “Sublinear Time Algorithms for Metric Space Problems”.) Let Dij denote the distance between a pair of points i and j, over m total points. The entries of Dij must satisfy the triangle inequality; additionally the matrix is symmetric. Note that the matrix size (i.e., ...

In this paper we consider bi-criteria geometric optimization problems, in particular, the minimum diameter minimum cost spanning tree problem and the minimum radius minimum cost spanning tree problem for a set of points in the plane. The former problem is to construct a minimum diameter spanning tree among all possible minimum cost spanning trees, while the latter is to construct a minimum radi...

We prove that every connected graph G of order n has a spanning tree T such that for every edge e of T the edge-cut defined in G by the vertex sets of the two components of T − e contains at most n 32 many edges which solves a problem posed by Ostrovskii (Minimal congestion trees, Discrete Math. 285 (2004), 219-226.)

An inverse spanning tree problem is to make the least modification on the edge weights such that a predetermined spanning tree is a minimum spanning tree. In this paper, based on the notion of fuzzy α-minimum spanning tree, the inverse spanning tree problem with fuzzy edge weights is discussed and formulated as a fuzzy programming model with some chance constraints. It shows that when all the e...

A k-tree is either a complete graph on k vertices or a graph T that contains a vertex whose neighbourhood in T induces a complete graph on k vertices and whose removal results in a k-tree. A subgraph of a graph is a spanning k-tree if it is a k-tree and contains every vertex of the graph. This paper is concerned with spanning 2-trees in a graph. It is shown that spanning 2-trees have close conn...

Proof. If the edge (r, s) from the root to the sink is included in the spanning tree, then each of the principal branches of Tn may be assigned an oriented spanning tree independently, so there are t n−1 such spanning trees. On the other hand, if (r, s) is not included in the spanning tree, there is a directed path r → x1 → . . . → xn−1 → s in the spanning tree from the root to the sink. In thi...

Spanning Tree Protocol (STP) is a link management standard that provides loop free paths in Ethernet networks. Deploying STP in metro area networks is inadequate because it does not meet the requirements of these networks. STP blocks redundant links, causing the risk of congestion close to the root. As a result, STP provides poor support for load balancing in metro Ethernet networks. A solution...