@MISC{Zelinka_findinga, author = {Bohdan Zelinka and Bohdan Zelinka}, title = {Finding a Spanning Tree of a Graph with Maximal Number of Terminal Vertices}, year = {} }

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Abstract

In [1] V. G. Vizing proposes the problem of finding an algorithm for finding a spanning tree of a given finite graph which would have the maximal number of terminal vertices. Here we shall give such an algorithm. At first we shall define some concepts and prove some theorems. We shall always consider finite undirected graphs without loops and multiple edges. The given problem is closely connected with the problem of finding the most economical communication network. If some spanning tree has the maximal number of. terminal vertices, it has the minimal number of the vertices of degree greater than one; this is a relevant re-quirement for the simplicity of a communication network in the sense of practical applications. Let G be a graph with the vertex set V. For each set S < = Vwe shall define the neigh-bourhood N(S) of S as the subset of V- S consisting of vertices which are joined with at least one vertex of S. The cardinality of the neighbourhood N(S) of S is called the degree of S and denoted by Q(S). It follows from this definition that Q(<J>) = Q(V) = = 0 and if S is a one-element set {u}, where ueV, then Q(S) is equal to the degree