{"paper":{"title":"Exponential Independence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dieter Rautenbach, Simon J\\\"ager","submitted_at":"2016-05-19T15:14:52Z","abstract_excerpt":"For a set $S$ of vertices of a graph $G$, a vertex $u$ in $V(G)\\setminus S$, and a vertex $v$ in $S$, let ${\\rm dist}_{(G,S)}(u,v)$ be the distance of $u$ and $v$ in the graph $G-(S\\setminus \\{ v\\})$. Dankelmann et al. (Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883) define $S$ to be an exponential dominating set of $G$ if $w_{(G,S)}(u)\\geq 1$ for every vertex $u$ in $V(G)\\setminus S$, where $w_{(G,S)}(u)=\\sum\\limits_{v\\in S}\\left(\\frac{1}{2}\\right)^{{\\rm dist}_{(G,S)}(u,v)-1}$. Inspired by this notion, we define $S$ to be an exponential independent set of $G$ if $w_{(G"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.05991","kind":"arxiv","version":1},"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"}