Showing papers on "Spanning tree published in 1968"

Kruskal's theorem for matroids

Abstract: Kruskal's theorem for obtaining a minimal (maximal) spanning tree of a graph is shown to be a special case of a more general theorem for matroid spaces in which each element of the matroid has an associated weight. Since any finite subset of a vector space can be regarded as a matroid space this theorem gives an easy method of selecting a linearly independent set of vectors of minimal (maximal) weight.

Spanning Tree Manipulation and the Travelling Salesman Problem

Abstract: The reasonable assumption is made that the majority of lines appearing in a minimal spanning tree for any network also appear in an optimal solution to the corresponding travelling salesman problem. A technique is described of manipulating the tree, by means of deletions and additions of lines, into a chain and hence obtain a feasible solution. An extension is considered with regard to the minimal wiring problem.

On Hamilton Circuits in Tree Graphs

Abstract: Some topological features of a tree graph are investigated. The results provide the decomposition of a tree graph into complete subgraphs. As an immediate consequence of the decomposition, two procedures for generating a Hamilton circuit in a tree graph are presented.

Minimal spanning trees with degree restraints

Abstract: i The purpose of t h i s t h e s i s i s t o develop a s o l u t i o n t o the problem of determining the minimal spanning tree with degree r e s t r a i n t s f o r a given non-directional graph. Section 1 gives an introduction t o the problem. A set of d e f i n i t i o n s describing the graphical terminology used i n the body of the t h e s i s , i s presented along with a d e s c r i p t i o n of the problem. At the end of t h i s section a few applications of the problem are given. Section 2 ou t l i n e s the method of so l u t i o n used. The algorithm incorporates a branch and bound technique and t h i s problem solving method i s discussed i n general i n the f i r s t part of the section. Some other app l i c a t i o n s of branching and bounding are also discussed. Next, the complete algorithm i s described along with a proof of optimality. A sample problem i s worked through t o i l l u s t r a t e the method of s o l u t i o n . Two d i f f e r e n t minimal spanning tree algorithms, one by R.C. Prim, the other by J.B. Kruskal, are used i n the main core of the s o l u t i o n algorithm. These two approaches are discussed with the a i d of a sample problem, at the end of Section 2. Computer programs were written t o t e s t the algorithms. Several sets of data were compiled f o r various s i z e s of graphs and values of degree r e s t r i c t i o n s . The r e s u l t s of these runs were tabulated and are discussed i n Section 3 . Next, a comparison i s made of the method discussed here and a so l u t i o n involving l i n e a r programming. Section 3 also presents some u s e f u l h e u r i s t i c approaches at suboptimization which e f f e c t i v e l y reduce the amount of computation. Section 4 summarizes the r e s u l t s of Section 3 and in d i c a t e s the best approach t o use f o r a s p e c i f i c problem. TABLE OF CONTENTS i i Section