{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T22:40:01Z","timestamp":1751064001906,"version":"3.41.0"},"reference-count":0,"publisher":"Combinatorial Press","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ars Comb."],"published-print":{"date-parts":[[2025,6,28]]},"abstract":"<jats:p>We consider the eternal distance-2 domination problem, recently proposed by Cox, Meger, and Messinger, on trees. We show that finding a minimum eternal distance-2 dominating set of a tree is linear time in the order of the graph by providing a fast algorithm. Additionally, we characterize  trees that have eternal distance-2 domination number equal to their domination number or their distance-2 domination number,  {along with trees that are} eternal distance-2 domination critical. We conclude by providing general upper and lower bounds for the eternal distance-k domination number of a graph. We construct an infinite family of trees which meet said upper bound and another infinite family of trees whose eternal distance-k domination number is within a factor of 2 of the given lower bound.<\/jats:p>","DOI":"10.61091\/ars163-03","type":"journal-article","created":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T22:01:43Z","timestamp":1751061703000},"page":"29-49","source":"Crossref","is-referenced-by-count":0,"title":["Eternal distance-2 domination in trees"],"prefix":"10.61091","volume":"163","author":[{"name":"Department of Mathematics, Simon Fraser University, Burnaby, Canada","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexander","family":"Clow","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christopher van","family":"Bommel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"name":"Department of Mathematics and Statistics, University of Guelph, Guelph, Canada","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"39747","published-online":{"date-parts":[[2025,6,28]]},"container-title":["Ars Combinatoria"],"original-title":[],"deposited":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T22:01:45Z","timestamp":1751061705000},"score":1,"resource":{"primary":{"URL":"https:\/\/combinatorialpress.com\/ars-articles\/volume-163\/eternal-distance-2-domination-in-trees\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,28]]},"references-count":0,"URL":"https:\/\/doi.org\/10.61091\/ars163-03","relation":{},"ISSN":["0381-7032","2817-5204"],"issn-type":[{"value":"0381-7032","type":"print"},{"value":"2817-5204","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,28]]}}}