Uncoverings on graphs and network reliability

TL;DR

This paper introduces a new graph-based network protocol using $t$-uncoverings-by-bases, which are collections of spanning trees ensuring coverage of edge subsets, with constructions for certain graph families and a conjecture on minimal size.

Contribution

It defines $t$-uncoverings-by-bases for graphs, constructs examples for specific graph families, and conjectures minimal size properties, extending the $k$-tree protocol framework.

Findings

Constructed $t$-uncoverings-by-bases for infinite graph families

Utilized graph factorizations and decompositions in constructions

Conjecture that uncoverings can be as small as the number of edges

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.