{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:56Z","timestamp":1753893836230,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum cardinality of the set of vertices with positive weight at the end of the process. In this paper, we investigate random geometric graphs $\\mathcal{G}(n,r)$ with $n$ vertices distributed uniformally at random in $[0,\\sqrt{n}]^2$ and two vertices being adjacent if and only if their distance is at most $r$. We show that asymptotically almost surely $a_t(\\mathcal{G}(n,r)) = \\Theta( n \/ (r \\lg r)^2)$ for the whole range of $r=r_n \\ge 1$ such that $r \\lg r \\le \\sqrt{n}$. By monotonicity, asymptotically almost surely $a_t(\\mathcal{G}(n,r)) = \\Theta(n)$ if $r &lt; 1$, and $a_t(\\mathcal{G}(n,r)) = \\Theta(1)$ if $r \\lg r &gt; \\sqrt{n}$.<\/jats:p>","DOI":"10.37236\/6632","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T10:50:11Z","timestamp":1578653411000},"source":"Crossref","is-referenced-by-count":2,"title":["The Total Acquisition Number of Random Geometric Graphs"],"prefix":"10.37236","volume":"24","author":[{"given":"Ewa","family":"Infeld","sequence":"first","affiliation":[]},{"given":"Dieter","family":"Mitsche","sequence":"additional","affiliation":[]},{"given":"Pawe\u0142","family":"Pra\u0142at","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,8,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p31\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p31\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:50:12Z","timestamp":1579218612000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i3p31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,11]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/6632","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,8,11]]},"article-number":"P3.31"}}