  We study graphs on $n$ vertices which have $2n-2$ edges and no proper induced
subgraphs of minimum degree $3$. Erd\H{o}s, Faudree, Gy\'arf\'as, and Schelp
conjectured that such graphs always have cycles of lengths $3,4,5,\dots, C(n)$
for some function $C(n)$ tending to infinity. We disprove this conjecture,
resolve a related problem about leaf-to-leaf path lengths in trees, and
characterize graphs with $n$ vertices and $2n-2$ edges, containing no proper
subgraph of minimum degree $3$.
