{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:48Z","timestamp":1753893828860,"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 a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$.\r\nNon-separating planar graphs are closed under taking minors and are a subclass of planar graphs and a superclass of outerplanar graphs.\r\nIn this paper, we show that a graph is a non-separating planar graph if and only if it does not contain $K_1 \\cup K_4$ or $K_1 \\cup K_{2,3}$ or $K_{1,1,3}$ as a minor.\r\nFurthermore, we provide a structural characterisation of this class of graphs. More specifically, we show that any maximal non-separating planar graph is either an outerplanar graph or a wheel or it is a graph obtained from the disjoint union of two triangles by adding three vertex-disjoint paths between the two triangles.\r\nLastly, to demonstrate an application of non-separating planar graphs, we use the characterisation of non-separating planar graphs to prove that there are maximal linkless graphs with $3n-3$ edges. Thus, maximal linkless graphs can have significantly fewer edges than maximum linkless graphs; Sachs exhibited linkless graphs with $n$ vertices and $4n-10$ edges (the maximum possible) in 1983.<\/jats:p>","DOI":"10.37236\/8816","type":"journal-article","created":{"date-parts":[[2021,1,14]],"date-time":"2021-01-14T06:13:30Z","timestamp":1610604810000},"source":"Crossref","is-referenced-by-count":5,"title":["Non-Separating Planar Graphs"],"prefix":"10.37236","volume":"28","author":[{"given":"Hooman R.","family":"Dehkordi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Graham","family":"Farr","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2021,1,15]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i1p11\/8251","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i1p11\/8251","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,14]],"date-time":"2021-01-14T06:13:30Z","timestamp":1610604810000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i1p11"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,15]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1,14]]}},"URL":"https:\/\/doi.org\/10.37236\/8816","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2021,1,15]]},"article-number":"P1.11"}}