Cut Vertices in Random Planar Graphs

We denote by P(n, m) a graph chosen uniformly at random from the class of all vertex-labelled planar graphs on vertex set \(\left\{ 1, \ldots , n\right\} \) with \(m=m(n)\) edges. We determine the asymptotic number of cut vertices in P(n, m) in the sparse regime. For comparison, we also derive the asymptotic number of cut vertices in the Erdős-Rényi random graph G(n, m).