For t > 0 and g = 0, a vertex-weighted graph of total weight W is (t, g)-trimmable if it contains a vertex-induced subgraph of total weight at least (1-1/t)W and with no simple path of more than g edges. A family of graphs is trimmable if for every constant t > 0, there is a constant g = 0 such that every vertex-weighted graph in the family is (t, g)-trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. We also show that every family of directed graphs of bounded layer bandwidth (a less restrictive condition than bounded directed bandwidth) is trimmable. As an application of these results, we derive polynomial-time approximation schemes for various forms of the problem of labeling a subset of given weighted point features with nonoverlapping sliding axes-parallel rectangular labels so as to maximize the total weight of the labeled features, provided that the ratios of label heights or the ratios of label lengths are bounded by a constant. This settles one of the last major open questions in the theory of map labeling. Keywords: Trimming weighted graphs - Domino treewidth - Planar graphs - Layer bandwidth - Point-feature label placement - Map labeling - Sliding labels - Polynomial-time approximation schemes.