Given a weighted undirected graph G=(V,E) with positive costs on the edges and a positive constant d, the min-degree constrained minimum spanning tree problem is to find a minimum cost spanning tree T of G where each node I of T either has degree at least d or is a leaf node. Thus, the goal is to find a spanning tree T where all its non-leaf nodes (or central nodes) have degree at least d. Thes...

This paper considers a generalized fuzzy random version of bottleneck spanning tree problem in which edge costs are fuzzy random variables. The problem is to find an optimal spanning tree under chance constraint with respect to possibility measure of bottleneck (maximum cost) edge of spanning tree. The problem is first transformed into a deterministic equivalent problem. Then its subproblem is ...

Spanning tree contours, a special class of Truchet contour based upon a random spanning tree of a Truchet tiling’s underlying graph, are presented. This spanning tree method is extended to three dimensions to define a Truchet surface with properties similar to its twodimensional counterpart. Both contour and surface are smooth, have known minimum curvature and known maximum distance to interior...

We consider the problem of finding a spanning tree satisfying a family of additional constraints. Several settings have been considered previously, the most famous being the problem of finding a spanning tree with degree constraints. Since the problem is hard, the goal is typically to find a spanning tree that violates the constraints as little as possible. Iterative rounding has become the too...

The first problem considered in this paper: is it possible to find upper estimates for the spanning tree congestion for bipartite graphs which are better than for general graphs? It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n 3 2 , where n is the number of vertices. The second problem is to estimate spanning tree congestion of rand...

The paper is devoted to estimates of the spanning tree congestion for some planar graphs. The main results of the paper: (1) We almost determined (up to ±1) the maximal possible spanning tree congestion for planar graphs. (2) The value of congestion indicator introduced in Ostrovskii [Discrete Math. 310 (2010), no. 6–7, 1204–1209] can be very far from the value of the spanning tree congestion. ...

Let G =< V,E > be a connected graph with real-valued edge weights: w : E → R, having n vertices and m edges. A spanning tree in G is an acyclic subgraph of G that includes every vertex of G and is connected; every spanning tree has exactly n− 1 edges. A minimum spanning tree (MST) is a spanning tree of minimum weight which is defined to be the sum of the weights of all its edges. Our problem is...

In their seminal paper on Euclidean minimum spanning trees [Discrete & Computational Geometry, 1992], Monma and Suri proved that any tree of maximum degree 5 admits a planar embedding as a Euclidean minimum spanning tree. Their algorithm constructs embeddings with exponential area; however, the authors conjectured that c × c area is sometimes required to embed an n-vertex tree of maximum degree...

We consider the well studied Full Degree Spanning Tree problem, a NP-complete variant of the Spanning Tree problem, in the realm of moderately exponential time exact algorithms. In this problem, given a graph G, the objective is to find a spanning tree T of G which maximizes the number of vertices that have the same degree in T as in G. This problem is motivated by its application in fluid netw...

Suppose G is a connected graph and T a spanning tree of G. A vertex v ∈ V (G) is said to be a degree-preserving vertex if its degree in T is the same as in G. The degree-preserving spanning tree problem is to find a spanning tree T of a connected graph G such that the number of degreepreserving vertices is maximum. The purpose of this paper is to give an O(mα(m,n))-time algorithm for the degree...