{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:05Z","timestamp":1753893845570,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Consider a graph each of whose vertices is either in the ON state or in the OFF state and call  the resulting ordered bipartition into ON vertices and OFF vertices a configuration of the graph.   A regular move at a vertex changes the states of the neighbors of that vertex and hence sends the current configuration to another one. A valid move is a regular move at an ON vertex.   For any graph $G,$ let $\\mathcal{D}(G)$   be the minimum integer such that given any starting configuration $\\bf x$ of $G$ there must exist a sequence of valid moves which takes $\\bf x$    to a configuration with at most $\\ell +\\mathcal{D}(G)$ ON vertices     provided there is a sequence of regular moves which brings $\\bf x$ to a  configuration in which there are $\\ell$ ON vertices.  The shadow graph $\\mathcal{S}(G)$ of a graph $G$ is obtained from $G$ by deleting all loops. We prove that  $\\mathcal{D}(G)\\leq 3$ if $\\mathcal{S}(G)$ is unicyclic and give an example to show that the bound $3$ is tight. We also prove that $\\mathcal{D}(G)\\leq 2$ if $ G $ is a two-dimensional grid graph and  $\\mathcal{D}(G)=0$ if $\\mathcal{S}(G)$ is a two-dimensional grid graph but not a path  and $G\\neq  \\mathcal{S}(G)$.<\/jats:p>","DOI":"10.37236\/701","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:39:52Z","timestamp":1578713992000},"source":"Crossref","is-referenced-by-count":1,"title":["Minimum Light Numbers in the $\\sigma$-Game and Lit-Only $\\sigma$-Game on Unicyclic and Grid Graphs"],"prefix":"10.37236","volume":"18","author":[{"given":"John","family":"Goldwasser","sequence":"first","affiliation":[]},{"given":"Xinmao","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Yaokun","family":"Wu","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2011,10,31]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p214\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p214\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:00:16Z","timestamp":1579302016000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p214"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10,31]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/701","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,10,31]]},"article-number":"P214"}}