Uncoverings on graphs and network reliability

Abstract

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.