{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:24:41Z","timestamp":1759335881931,"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>Given a graph $G$, a labeling $c:V(G) \\rightarrow \\{1, 2, \\ldots, d\\}$ is said to be $d$-distinguishing if the only element in ${\\rm Aut}(G)$ that preserves the labels is the identity.  The distinguishing number of $G$, denoted by $D(G)$, is the minimum $d$ such that $G$ has a $d$-distinguishing labeling. If $G \\square H$ denotes the Cartesian product of $G$ and $H$, let $G^{^2} = G \\square G$ and $G^{^r} = G \\square G^{^{r-1}}$. A graph $G$ is said to be prime with respect to the Cartesian product if whenever $G \\cong G_1 \\square G_2$, then either $G_1$ or $G_2$ is a singleton vertex.  This paper proves that if $G$ is a connected, prime graph, then $D(G^{^r}) = 2$ whenever $r \\geq 4$.<\/jats:p>","DOI":"10.37236\/1984","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:09:33Z","timestamp":1578701373000},"source":"Crossref","is-referenced-by-count":23,"title":["Distinguishing Cartesian Powers of Graphs"],"prefix":"10.37236","volume":"12","author":[{"given":"Michael O.","family":"Albertson","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2005,9,19]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1n17\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1n17\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:40:56Z","timestamp":1579304456000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v12i1n17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,9,19]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2005,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1984","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2005,9,19]]},"article-number":"N17"}}