For a graph G, let f(G) denote the maximum number of edges in a cut of G. For an integer m and for a fixed graph H, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$, as G ranges over all graphs on m edges that contain no copy of H. In this paper we study this function for various graphs H. In particular we show that for any graph H obtained by connecting a single vertex to all vertices of a fixed nontrivial forest, there is a $c(H) >0$ such that $f(m,H) \geq \frac{m}{2} + c(H) m^{4/5}$, and that this is tight up to the value of $c(H)$. We also prove that for any even cycle $C_{2k}$ there is a $c(k)>0$ such that $f(m,C_{2k}) \geq \frac{m}{2} + c(k) m^{(2k+1)/(2k+2)}$, and that this is tight, up to the value of $c(k)$, for $2k\in \{4,6,10\}$. The proofs combine combinatorial, probabilistic and spectral techniques.