Cut Vertices in Random Planar Maps

Abstract

The main goal of this paper is to determine the asymptotic behavior of the number $X_n$ of cut-vertices in random planar maps with $n$ edges. It is shown that $X_n/n\to c$ in probability (for some explicit $c>0$). For so-called subcritical classes of planar maps (like outerplanar maps) we obtain a central limit theorem, too. Interestingly the combinatorics behind this seemingly simple problem is quite involved.