  For $t,g>0$, 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 each constant $t>0$, there is a constant $g=g(t)$ 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. Based on this result, we derive a
polynomial-time approximation scheme for the problem of labeling weighted
points with nonoverlapping sliding labels of unit height and given lengths so
as to maximize the total weight of the labeled points. This settles one of the
last major open questions in the theory of map labeling.
