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On the steiner problem Cockayne, Ernest

Abstract

The classical Steiner Problem may be stated: Given n points [formula omitted] in the Euclidean plane, to construct the shortest tree(s) (i.e. undirected, connected, circuit free graph(s)) whose vertices include [formula omitted]. The problem is generalised by considering sets in a metric space rather than points in E² and also by minimising a more general graph function than length, thus yielding a large class of network minimisation problems which have a wide variety of practical applications, The thesis is concerned with the following aspects of these problems. 1. Existence and uniqueness or multiplicity of solutions. 2. The structure of solutions and demonstration that minimising trees of various problems share common properties. 3. Solvability of problems by Euclidean constructions or by other geometrical methods.

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