{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:50:18Z","timestamp":1759063818086,"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 vertex $v$ of a graph $G$ is a boundary vertex if there exists a vertex $u$ such that the distance in $G$ from $u$ to $v$ is at least the distance from $u$ to any neighbour of $v$.  We give a full description of all graphs that have exactly four boundary vertices, which answers a question of Hasegawa and Saito.  To this end, we introduce the concept of frame of a graph. It allows us to construct, for every positive integer $b$ and every possible \"distance-vector\" between $b$ points, a graph $G$ with exactly $b$ boundary vertices such that every graph with $b$ boundary vertices and the same distance-vector between them is an induced subgraph of $G$.<\/jats:p>","DOI":"10.37236\/498","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:57:16Z","timestamp":1578715036000},"source":"Crossref","is-referenced-by-count":3,"title":["Graphs with Four Boundary Vertices"],"prefix":"10.37236","volume":"18","author":[{"given":"Tobias","family":"M\u00fcller","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Attila","family":"P\u00f3r","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean-S\u00e9bastien","family":"Sereni","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,1,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p11\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p11\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:18:09Z","timestamp":1579303089000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p11"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1,5]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/498","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,1,5]]},"article-number":"P11"}}