AbstractLet G be an infinite connected graph with minimum degree δ and maximum degree Δ. Let Gp be a random induced subgraph of G obtained by selecting each vertex of G independently with probability p, , and let be the induced subgraph of Gp obtained by deleting all vertices of Gp with degree greater than k in Gp. We show that if and is not too large then almost surely has no infinite component. Moreover, this result is essentially best possible since there are examples where has an infinite component (a) when , 4, or 5, and k = 3; (b) when for any δ and k = 3; and (c) when for any and . In addition, we show that if G is the d‐dimensional lattice then almost surely has an infinite component for sufficiently large d. © 2014 Wiley Periodicals, Inc. Random Struct. Alg. 44, 399–418, 2014