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Let $G(n,\, M)$ be the uniform random graph with $n$ vertices and $M$ edges.Let $B_n$ be the maximum block-size of $G(n,\, M)$ or the maximum size of itsmaximal $2$-connected induced subgraphs. We determine the expectation of $B_n$near the critical point $M=n/2$. As $n-2M \gg n^{2/3}$, we find a constant$c_1$ such that \[ c_1 = \lim_{n \rightarrow \infty} \left(1 - \frac{2M}{n}\right) \, E B_n \, . \] Inside the window of transition of $G(n,\, M)$ with$M=\frac{n}{2}(1+\lambda n^{-1/3})$, where $\lambda$ is any real number, wefind an exact analytic expression for \[ c_2(\lambda) = \lim_{n \rightarrow\infty} \frac{E B_n} {n^{1/3}} \, . \] This study relies on the symbolic methodand analytic tools coming from generating function theory which enable us todescribe the evolution of $n^{-1/3} \, E B_n $ as a function of $\lambda$.

The Maximum Block Size of Critical Random Graphs