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Minimal vertex covers of random trees

Stephane Coulomb
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Abstract

We study minimal vertex covers of trees. Contrarily to the number $N_{vc}(A)$ of minimal vertex covers of the tree $A$, $\log N_{vc}(A)$ is a self-averaging quantity. We show that, for large sizes $n$, $\lim_{n\to +\infty} <\log N_{vc}(A)>_n/n= 0.1033252\pm 10^{-7}$. 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 $<\log N_{vc}>$ over totally degenerate trees of given size, which we perform numerically.

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Stephane Coulomb. 2004-11-15. Minimal vertex covers of random trees. https://doi.org/10.1088/1742-5468%2F2005%2F06%2Fp06007

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