{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T23:36:11Z","timestamp":1648596971042},"reference-count":7,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2011,9]]},"abstract":"<jats:p> For a connected graph G of order n, an ordered set S = {u<jats:sub>1<\/jats:sub>, u<jats:sub>2<\/jats:sub>, \u2026, u<jats:sub>k<\/jats:sub>} of vertices in G is a linear geodetic set of G if for each vertex x in G, there exists an index i, 1 \u2264 i &lt; k such that x lies on a u<jats:sub>i<\/jats:sub> - u<jats:sub>i + 1<\/jats:sub> geodesic on G, and a linear geodetic set of minimum cardinality is the linear geodetic number g<jats:sub>l<\/jats:sub>(G). The linear geodetic numbers of certain standard graphs are obtained. It is shown that if G is a graph of order n and diameter d, then g<jats:sub>l<\/jats:sub>(G) \u2264 n - d + 1 and this bound is sharp. For positive integers r, d and k \u2265 2 with r &lt; d \u2264 2r, there exists a connected graph G with rad G = r, diam G = d and g<jats:sub>l<\/jats:sub>(G) = k. Also, for integers n, d and k with 2 \u2264 d &lt; n, 2 \u2264 k \u2264 n - d + 1, there exists a connected graph G of order n, diameter d and g<jats:sub>l<\/jats:sub>(G) = k. We characterize connected graphs G of order n with g<jats:sub>l<\/jats:sub>(G) = n and g<jats:sub>l<\/jats:sub>(G) = n - 1. It is shown that for each pair a, b of integers with 3 \u2264 a \u2264 b, there is a connected graph G with g(G) = a and g<jats:sub>l<\/jats:sub>(G) = b. We also discuss how the linear geodetic number of a graph is affected by adding a pendent edge to the graph. <\/jats:p>","DOI":"10.1142\/s1793830911001279","type":"journal-article","created":{"date-parts":[[2011,10,21]],"date-time":"2011-10-21T12:29:23Z","timestamp":1319200163000},"page":"357-368","source":"Crossref","is-referenced-by-count":1,"title":["THE LINEAR GEODETIC NUMBER OF A GRAPH"],"prefix":"10.1142","volume":"03","author":[{"given":"A. P.","family":"SANTHAKUMARAN","sequence":"first","affiliation":[{"name":"Department of Mathematics, St. Xavier's College (Autonomous), Palayamkottai-627 002, India"}]},{"given":"T.","family":"JEBARAJ","sequence":"additional","affiliation":[{"name":"Department of Mathematics, C.S.I. Institute of Technology, Thovalai, India"}]},{"given":"S. V. ULLAS","family":"CHANDRAN","sequence":"additional","affiliation":[{"name":"Department of Mathematics, St. Xavier's College (Autonomous), Palayamkottai-627 002, India"}]}],"member":"219","published-online":{"date-parts":[[2012,4,5]]},"reference":[{"key":"rf1","volume-title":"Distance in Graphs","author":"Buckley F.","year":"1990"},{"key":"rf2","first-page":"51","volume":"31","author":"Chartrand G.","journal-title":"Bull. ICA"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1002\/net.10007"},{"key":"rf4","doi-asserted-by":"crossref","DOI":"10.21236\/AD0705364","volume-title":"Graph Theory","author":"Harary F.","year":"1969"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1016\/0895-7177(93)90259-2"},{"key":"rf6","first-page":"161","volume":"143","author":"Muntean R.","journal-title":"Congr. Numer."},{"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\/S1793830911001279","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T16:12:54Z","timestamp":1565194374000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1793830911001279"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,9]]},"references-count":7,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2012,4,5]]},"published-print":{"date-parts":[[2011,9]]}},"alternative-id":["10.1142\/S1793830911001279"],"URL":"https:\/\/doi.org\/10.1142\/s1793830911001279","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"value":"1793-8309","type":"print"},{"value":"1793-8317","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,9]]}}}