  We propose a network protocol similar to the $k$-tree protocol of Itai and
Rodeh [{\em Inform.\ and Comput.}\ {\bf 79} (1988), 43--59]. To do this, we
define an {\em $t$-uncovering-by-bases} for a connected graph $G$ to be a
collection $\mathcal{U}$ of spanning trees for $G$ such that any $t$-subset of
edges of $G$ is disjoint from at least one tree in $\mathcal{U}$, where $t$ is
some integer strictly less than the edge connectivity of $G$. We construct
examples of these for some infinite families of graphs. Many of these infinite
families utilise factorisations or decompositions of graphs. In every case the
size of the uncovering-by-bases is no larger than the number of edges in the
graph and we conjecture that this may be true in general.
