Abstract
We study minimal vertex covers of trees. In contrast to the numberNvc(A) of minimal vertexcovers of the tree A, logNvc(A) is a self-averaging quantity. We show that, for large sizesn, . The basic idea is, given a tree, to concentrate on its degenerate vertices, that is thosevertices which belong to some minimal vertex cover but not to all of them. Deletion ofthe other vertices induces a forest of totally degenerate trees. We show that theproblem reduces to the computation of the size distribution of this forest, which weperform analytically, and of the average over totally degenerate trees of given size, which we perform numerically.
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