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We prove that every connected graph $G$ with $m$ edges contains a set $X$ ofat most $\frac{3}{16}(m + 1)$ vertices such that $G-X$ has no $K_4$ minor, orequivalently, has treewidth at most $2$. This bound is best possible.Connectivity is essential: If $G$ is not connected then only a bound of$\frac{1}{5}m$ can be guaranteed.

$K_{4}$-Minor-Free Induced Subgraphs of Sparse Connected Graphs