{"paper":{"title":"$k$-Blocks: a connectivity invariant for graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fabian Hundertmark, Johannes Carmesin, Matthias Hamann, Reinhard Diestel","submitted_at":"2013-05-20T15:34:13Z","abstract_excerpt":"A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by fewer than $k$ other vertices. The block number $\\beta(G)$ of $G$ is the largest integer $k$ such that $G$ has a $k$-block.\n  We investigate how $\\beta$ interacts with density invariants of graphs, such as their minimum or average degree. We further present algorithms that decide whether a graph has a $k$-block, or which find all its $k$-blocks.\n  The connectivity invariant $\\beta(G)$ has a dual width invariant, the block-width ${\\rm bw}(G)$ of $G$. Our algorithms imply the duality t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4557","kind":"arxiv","version":4},"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"}