  Given an edge-weighted graph $G$ and $\epsilon>0$, a $(1+\epsilon)$-spanner
is a spanning subgraph $G'$ whose shortest path distances approximate those of
$G$ within a $(1+\epsilon)$ factor. If $G$ is from certain minor-closed graph
families (at least bounded genus graphs and apex graphs), then we know that
light spanners exist. That is, we can compute a $(1+\epsilon)$-spanner $G'$
with total edge weight at most a constant times the weight of a minimum
spanning tree. This constant may depend on $\epsilon$ and the graph family, but
not on the particular graph $G$ nor on its edge weighting. For weighted graphs
from several minor-closed graph families, the existence of light spanners has
been essential in the design of approximation schemes for the metric TSP (the
traveling salesman problem) and some similar problems. In this paper we make
some progress towards the conjecture that light spanners exist for every
minor-closed graph family. In particular, we show that they exist for graphs
with bounded pathwidth. We do this via the construction of light enough
monotone spanning trees in such graphs.
