{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:50:15Z","timestamp":1759063815345,"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>A graph $G$ is said to be $2$-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the label classes. \u00a0The minimum size of a label class in any such labeling of $G$ is called the cost of $2$-distinguishing $G$ and is denoted by $\\rho(G)$. \u00a0The determining number of a graph $G$, denoted $\\det(G)$, is the minimum size of a set of vertices whose pointwise stabilizer is trivial. \u00a0The main result of this paper is that if $G^k$ is a $2$-distinguishable Cartesian power of a prime, connected graph $G$ on at least three vertices with $\\det(G)\\leq k$ and $\\max\\{2, \\det(G)\\} &lt; \\det(G^k)$, then $\\rho(G^k) \\in \\{\\det(G^k), \\det(G^k)+1\\}$. \u00a0In particular, for $n\\geq 3$, $\\rho(K_3^n)\\in \\{ \\left\\lceil {\\log_3 (2n+1)} \\right\\rceil$ $+1, \\left\\lceil {\\log_3 (2n+1)} \\right\\rceil$ $+2\\}$.<\/jats:p>","DOI":"10.37236\/3223","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T07:02:00Z","timestamp":1578726120000},"source":"Crossref","is-referenced-by-count":4,"title":["The Cost of 2-Distinguishing Cartesian Powers"],"prefix":"10.37236","volume":"20","author":[{"given":"Debra","family":"Boutin","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,3,31]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i1p74\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i1p74\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:23:15Z","timestamp":1579260195000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i1p74"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,3,31]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2013,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/3223","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,3,31]]},"article-number":"P74"}}