A Few Remarks on the History of Mst{problem Dedicated to the Memory of Professor Otakar Bor Uvka

On the background of Bor uvka's pioneering work we present a survey of the development related to the Minimum Spanning Tree Problem. We also complement the historical paper Graham-Hell GH] by a few remarks and provide an update of the extensive literature devoted to this problem. In the contemporary terminology the Minimum Spanning Tree problem can be formulated as follows: Given a nite set V and a real weight function w on pairs of elements of V nd a tree (V; T) of minimal weight w(t) = P ? w(x; y) : fx; yg 2 T. For example when V is a subset of a metric space and the weight function is deened as the distance then a solution T presents the shortest network connecting all the points V. Another frequent formulation which also explains its name is MST PROBLEM : Given a connected (undirected) graph G = (V; E) with real weights attached to its edges nd a spanning tree (V; T) of G (i.e. T E) such that the total weight w(T) is minimal. This is a cornerstone problem of Combinatorial Optimization and in a sense its cradle. The problem is important both in its practical and theoretical applications. We want to demonstrate this interest seems not to be dying until now. The problem was isolated and attacked in the fties with the vigor and conn-dence of then newly developing elds theory of algorithms and computer science. The contributions were numerous and illustrious: K. (see the references : it is only tting and fortunate that this Bor uvka's memorial volume contains a reminiscence of these early days written by J. B. Kruskal). These pioneering works made the MST problem popular and the further development only contributed to it. The paper of R. L. Graham and P. Hell GH] described accurately the development until 1985. Here are some of the main features that indicate the role and importance of this problem in contemporary discrete mathematics:

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