{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T16:37:04Z","timestamp":1774370224998,"version":"3.50.1"},"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 labeling of the vertices of a graph G, $\\phi :V(G) \\rightarrow \\{1,\\ldots,r\\}$, is said to be $r$-distinguishing provided no automorphism of the graph preserves all of the vertex labels. The distinguishing number of a graph G, denoted by $D(G)$, is the minimum $r$ such that $G$ has an $r$-distinguishing labeling. The distinguishing number of the complete graph on $t$ vertices is $t$.  In contrast, we prove (i) given any group $\\Gamma$, there is a graph $G$ such that $Aut(G) \\cong \\Gamma$ and $D(G)= 2$;  (ii) $D(G) = O(log(|Aut(G)|))$; (iii) if $Aut(G)$ is abelian, then $D(G) \\leq 2$;  (iv) if $Aut(G)$ is dihedral, then  $D(G) \\leq 3$;  and (v) If $Aut(G) \\cong S_4$, then either $D(G) = 2$ or $D(G) = 4$.  Mathematics Subject Classification 05C,20B,20F,68R<\/jats:p>","DOI":"10.37236\/1242","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:26:32Z","timestamp":1578687992000},"source":"Crossref","is-referenced-by-count":107,"title":["Symmetry Breaking in Graphs"],"prefix":"10.37236","volume":"3","author":[{"given":"Michael O.","family":"Albertson","sequence":"first","affiliation":[]},{"given":"Karen L.","family":"Collins","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[1996,6,9]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v3i1r18\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v3i1r18\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T01:17:23Z","timestamp":1579310243000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v3i1r18"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,6,9]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[1996,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1242","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,6,9]]},"article-number":"R18"}}