Minimal vertex covers of random trees
Stephane Coulomb (SPhT)

TL;DR
This paper investigates the properties of minimal vertex covers in large random trees, revealing that the logarithm of the number of such covers is a self-averaging quantity with a specific growth rate.
Contribution
It provides an analytical and numerical study of the distribution and growth rate of minimal vertex covers in large random trees, focusing on degenerate vertices.
Findings
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Abstract
We study minimal vertex covers of trees. Contrarily to the number of minimal vertex covers of the tree , is a self-averaging quantity. We show that, for large sizes , . The basic idea is, given a tree, to concentrate on its degenerate vertices, that is those vertices which belong to some minimal vertex cover but not to all of them. Deletion of the other vertices induces a forest of totally degenerate trees. We show that the problem reduces to the computation of the size distribution of this forest, which we perform analytically, and of the average over totally degenerate trees of given size, which we perform numerically.
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