{"paper":{"title":"Uncoverings on graphs and network reliability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brett Stevens, Robert F. Bailey","submitted_at":"2010-12-14T04:40:40Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2928","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}