a {em 2-rainbow dominating function} (2rdf) of a graph $g$ is a function $f$ from the vertex set $v(g)$ to the set of all subsets of the set ${1,2}$ such that for any vertex $vin v(g)$ with $f(v)=emptyset$ the condition $bigcup_{uin n(v)}f(u)={1,2}$ is fulfilled, where $n(v)$ is the open neighborhood of $v$. the {em weight} of a 2rdf $f$ is the value $omega(f)=sum_{vin v}|f (v)|$. the {em $2$-r...

A tree t-spanner of an unweighted graph G is a spanning tree T such that for every two vertices their distance in T is at most t times their distance in G. Given an unweighted graph G and a positive integer t as input, the tree t-spanner problem is to compute a tree t-spanner of G if one exists. This decision problem is known to be NP-complete even in the restricted class of unweighted planar g...

In this paper, we formulate an optimization problem for the design of light-tree based logical topology in Wavelength Division Multiplexing (WDM) networks. The problem is comprised of two parts: (1) multicast routing and wavelength assignment of light-trees, and (2) the design of light-tree based logical topology for multicast streams. In the first part, we use Mixed Integer Linear Programming ...

In this paper we initiate a systematic study of the Reduced Degree Spanning Tree problem, where given a digraph D and a nonnegative integer k, the goal is to construct a spanning out-tree with at most k vertices of reduced out-degree. This problem is a directed analog of the wellstudied Minimum-Vertex Feedback Edge Set problem. We show that this problem is fixed-parameter tractable and admits a...

The Diameter-Constrained Minimum Spanning Tree Problem seeks a least cost spanning tree subject to a (diameter) bound imposed on the number of edges in the tree between any node pair. A traditional multicommodity flow model with a commodity for every pair of nodes was unable to solve a 20-node and 100-edge problem after one week of computation. We formulate the problem as a directed tree from a...

Let T be a tree. A vertex of T with degree one is called a leaf, and the set of leaves of T is denoted by Leaf(T ). The subtree T − Leaf(T ) of T is called the stem of T and denoted by Stem(T ). A spanning tree with specified stem was first considered in [3]. A tree whose maximum degree at most k is called a k-tree. Similarly, a stem whose maximum degree at most k in it is called a k-stem, and ...

An enumerative scheme to solve this problem was proposed by Smith [11]. Maculan, Michelon and Xavier [7] formulated the ESTP as a nonconvex mixed-integer programming problem and proposed a Lagrangean dual program in order to develop a branch-and-bound method. A number of heuristic approaches have been developed in dimension n = 2 (see [13] for a survey). Heuristics for n ≥ 3 can be found in [6,...

Let p be a graph parameter that assigns a positive integer value to every graph. The inverse problem for p asks for a graph within a prescribed class (here, we will only be concerned with trees), given the value of p. In this context, it is of interest to know whether such a graph can be found for all or at least almost all integer values of p. We will provide a very general setting for this ty...

We study streaming algorithms for partitioning integer sequences and trees. In the case of trees, we suppose that the input tree is provided by a stream consisting of a depth-first-traversal of the input tree. This captures the problem of partitioning XML streams, among other problems. We show that both problems admit deterministic (1+ )-approximation streaming algorithms, where a single pass i...

Imagine a tree with some integer amount of gold at each vertex. Two players can play a game by taking turns removing leaves one by one and taking the gold from those leaves. We prove a recent conjecture of Micek and Walczak that says that if a tree has an even number of vertices, the first player can always secure at least half of the gold.