Trimming weighted graphs of bounded treewidth

Erlebach et al. define the notion of a trimmable class of graphs. This was motivated by a problem in map labelling. Roughly speaking, a class CC of graphs is trimmable if, for each graph G∈CG∈C, it is possible to remove a small proportion of the vertices of GG to obtain a graph with no long simple paths. More generally, one considers vertex-weighted graphs and tries to remove a set of vertices of small weight. Erlebach et al. prove that any class of weighted graphs of bounded treewidth and bounded degree is trimmable. They ask whether this remains true if the degree is not required to be bounded. In this paper a positive answer is given.

► Every weighted small-treewidth graph GG has a small vertex set meeting all long paths.
► The proof uses a (tw(G)+1)(tw(G)+1)-colouring of GG and recursion on the colours.
► It is known that this answers a question about map labelling.