{"paper":{"title":"Characterizing Block Graphs in Terms of their Vertex-Induced Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. Dress, A. Spillner, J. Koolen, K. T. Huber, V. Moulton","submitted_at":"2014-02-18T10:31:59Z","abstract_excerpt":"Given a finite connected simple graph $G=(V,E)$ with vertex set $V$ and edge set $E\\subseteq \\binom{V}{2}$, we will show that\n  $1.$ the (necessarily unique) smallest block graph with vertex set $V$ whose edge set contains $E$ is uniquely determined by the $V$-indexed family ${\\bf P}_G:=\\big(\\pi_0(G^{(v)})\\big)_{v \\in V}$ of the various partitions $\\pi_0(G^{(v)})$ of the set $V$ into the set of connected components of the graph $G^{(v)}:=(V,\\{e\\in E: v\\notin e\\})$,\n  $2.$ the edge set of this block graph coincides with set of all $2$-subsets $\\{u,v\\}$ of $V$ for which $u$ and $v$ are, for all "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.4277","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"}