  We study the Karp--Sipser core of a random graph made of a configuration
model with vertices of degree $1,2$ and $3$. This core is obtained by
recursively removing the leaves as well as their unique neighbors in the graph.
We settle a conjecture of Bauer & Golinelli and prove that at criticality, the
Karp--Sipser core has size $ \approx \mathrm{Cst} \cdot \vartheta^{-2} \cdot
n^{3/5}$ where $\vartheta$ is the hitting time of the curve $t \mapsto
\frac{1}{t^{2}}$ by a linear Brownian motion started at $0$. Our proof relies
on a detailed multi-scale analysis of the Markov chain associated to
Karp-Sipser leaf-removal algorithm close to its extinction time.
