Before revision:  We propose a polynomial delay and exponential space enumeration algorithm of minimal induced Steiner subgraphs for claw-free graphs , whereas the problem on general graphs is shown to be at least as hard as the problem of enumerating minimal transversals in hypergraphs . Contrary to these tractable results, we show that the problem of enumerating minimal group Steiner trees is at least as hard as the minimal transversal enumeration problem on hypergraphs.
After revision:  We propose a polynomial delay and exponential space enumeration algorithm of minimal induced Steiner subgraphs on claw-free graphs , whereas the problem on general graphs is shown to be at least as hard as the problem of enumerating minimal transversals in hypergraphs . Contrary to these tractable results, we show that the problem of enumerating minimal group Steiner trees is at least as hard as the minimal transversal enumeration problem on hypergraphs.