A linear-time algorithm for the minimum-ratio spanning tree problem on planar graphs is presented. The algorithm is based on a new planar minimum spanning tree algorithm. The approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having small separators. Disciplines Theory and Algorithms This article is available at Iowa State Univer...

Minimum spanning tree theory has a wide application in many fields. But in many practical problems, we are often faced with the heterogeneous node weighted graph with both edge weight and node weight be considered. In this paper, we present the definition and the mathematical model of the best spanning tree, then raise an algorithm of the best spanning tree, finally, prove that the algorithm is...

We show that the spanning tree degree enumerator polynomial of a connected graph G is real stable if and only distance-hereditary.

We present two short proofs for Diestel's criterion that a connected graph has normal spanning tree provided it contains no subdivision of countable clique in which every edge been replaced by uncountably many parallel edges.

Inspired by the seminal works of Khuller et al. (SIAM J. Comput. 25(2), 355–368 (1996)) and Chan (Discrete Geom. 32(2), 177–194 (2004)) we study bottleneck version Euclidean bounded-degree spanning tree problem. A is a whose largest edge-length minimum, degree-K minimum. Let $$\beta _K$$ be supremum ratio to tree, over all finite point sets in plane. It known that _5=1$$ , it easy verify _2\geq...

The degree-constrained minimum spanning tree (DCMST) is relevant in the design of networks. It consists of finding a spanning tree whose nodes do not exceed a given maximum degree and whose total edge length is minimum. We design a primal branch-and-cut algorithm that solves instances of the problem to optimality. Primal methods have not been used extensively in the past, and their performance ...