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This article provides different approaches to certain models of graph theory, where the minimum spanning tree (MST) models are suitable. Graph theory knows a variety of methods how to solve this problem of looking for the minimum spanning tree and  this article compares two of them in terms of their choice of use. The principle of the MST problem describes various kinds of situations where it is necessary to use this theoretical instrument, to find how to use this method in finding a solution, and finally to compare two methods of looking for the MST, in terms of their different approaches, their complementarity, and their assessment, which of these two methods can find a feasible solution faster in particular cases. A theoretical discussion and a model example are carried out to compare the two methods.