{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T08:48:23Z","timestamp":1770454103511,"version":"3.49.0"},"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 a $k$-dot product graph if there exists a vector labelling $u: V(G) \\to \\mathbb{R}^k$ such that $u(i)^{T}u(j) \\geq 1$ if and only if $ij \\in E(G)$. Fiduccia, Scheinerman, Trenk and Zito [Discrete Math., 1998] asked whether every planar graph is a $3$-dot product graph. We show that the answer is \"no\". On the other hand, every planar graph is a $4$-dot product graph. We also answer the corresponding questions for planar graphs of prescribed girth and for outerplanar graphs.<\/jats:p>","DOI":"10.37236\/703","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:39:47Z","timestamp":1578713987000},"source":"Crossref","is-referenced-by-count":3,"title":["Dot Product Representations of Planar Graphs"],"prefix":"10.37236","volume":"18","author":[{"given":"Ross J.","family":"Kang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"L\u00e1szl\u00f3","family":"Lov\u00e1sz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tobias","family":"M\u00fcller","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Edward R.","family":"Scheinerman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,11,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p216\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p216\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:59:16Z","timestamp":1579301956000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p216"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,11,7]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/703","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,11,7]]},"article-number":"P216"}}