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

Uncoverings on graphs and network reliability