{"paper":{"title":"Two-lit trees for lit-only sigma-game","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hau-wen Huang","submitted_at":"2010-10-28T03:23:21Z","abstract_excerpt":"A configuration of the lit-only $\\sigma$-game on a finite graph $\\Gamma$ is an assignment of one of two states, on or off, to all vertices of $\\Gamma.$ Given a configuration, a move of the lit-only $\\sigma$-game on $\\Gamma$ allows the player to choose an on vertex $s$ of $\\Gamma$ and change the states of all neighbors of $s.$ Given any integer $k$, we say that $\\Gamma$ is $k$-lit if, for any configuration, the number of on vertices can be reduced to at most $k$ by a finite sequence of moves. Assume that $\\Gamma$ is a tree with a perfect matching. We show that $\\Gamma$ is 1-lit and any tree obt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.5846","kind":"arxiv","version":3},"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"}