{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T21:41:21Z","timestamp":1648762881113},"reference-count":7,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2012,6]]},"abstract":"<jats:p> For a connected graph G of order p \u2265 2 and a set W \u2286 V(G), a tree T contained in G is a Steiner tree with respect to W if T is a tree of minimum order with W \u2286 V(T). The set S(W) consists of all vertices in G that lie on some Steiner tree with respect to W. The set W is a Steiner set for G if S(W) = V(G). The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. A geodetic set of G is a set S of vertices such that every vertex of G is contained in a geodesic joining some pair of vertices of S. The geodetic number g(G) of G is the minimum cardinality of its geodetic sets and any geodetic set of cardinality g(G) is a minimum geodetic set of G. A vertex v is an extreme vertex of a graph G if the subgraph induced by its neighbors is complete. The number of extreme vertices in G is its extreme order ex (G). A graph G is an extreme Steiner graph if s(G) = ex (G), and an extreme geodesic graph if g(G) = ex (G). Extreme Steiner graphs of order p with Steiner number p - 1 are characterized. It is shown that every pair a, b of integers with 0 \u2264 a \u2264 b is realizable as the extreme order and Steiner number, respectively, of some graph. For positive integers r, d and l \u2265 2 with r &lt; d \u2264 2r, it is shown that there exists an extreme Steiner graph G of radius r, diameter d, and Steiner number l. For integers p, d and k with 2 \u2264 d &lt; p, 2 \u2264 k &lt; p and p - d - k + 2 \u2265 0, there exists an extreme Steiner graph G of order p, diameter d and Steiner number k. It is shown that for every pair a, b of integers with 3 \u2264 a &lt; b and b = a + 1, there exists an extreme Steiner graph G with s(G) = a and g(G) = b that is not an extreme geodesic graph. It is shown that for every pair a, b of integers with 3 \u2264 a &lt; b, there exists an extreme geodesic graph G with g(G) = a and s(G) = b that is not an extreme Steiner graph. <\/jats:p>","DOI":"10.1142\/s1793830912500292","type":"journal-article","created":{"date-parts":[[2012,6,19]],"date-time":"2012-06-19T14:55:53Z","timestamp":1340117753000},"page":"1250029","source":"Crossref","is-referenced-by-count":1,"title":["EXTREME STEINER GRAPHS"],"prefix":"10.1142","volume":"04","author":[{"given":"A. P.","family":"SANTHAKUMARAN","sequence":"first","affiliation":[{"name":"Research Department of Mathematics, St. Xavier's College (Autonomous), Palayamkottai - 627 002, India"}]}],"member":"219","published-online":{"date-parts":[[2012,6,21]]},"reference":[{"key":"rf1","volume-title":"Distance in Graphs","author":"Buckley F.","year":"1990"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1002\/net.10007"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(00)00456-8"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1023\/B:CMAJ.0000027232.97642.45"},{"key":"rf5","doi-asserted-by":"crossref","DOI":"10.21236\/AD0705364","volume-title":"Graph Theory","author":"Harary F.","year":"1969"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1016\/0895-7177(93)90259-2"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(73)90116-7"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830912500292","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T22:42:21Z","timestamp":1565131341000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1793830912500292"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,6]]},"references-count":7,"journal-issue":{"issue":"02","published-online":{"date-parts":[[2012,6,21]]},"published-print":{"date-parts":[[2012,6]]}},"alternative-id":["10.1142\/S1793830912500292"],"URL":"https:\/\/doi.org\/10.1142\/s1793830912500292","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"value":"1793-8309","type":"print"},{"value":"1793-8317","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,6]]}}}