{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:14:19Z","timestamp":1758824059478,"version":"3.37.3"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,1,7]],"date-time":"2019-01-07T00:00:00Z","timestamp":1546819200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"name":"S.K. acknowledges the financial support from the Slovenian Research Agency","award":["research core funding No. P1-0297 and projects J1-7110, J1-9109, N1-0043"],"award-info":[{"award-number":["research core funding No. P1-0297 and projects J1-7110, J1-9109, N1-0043"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cent Eur J Oper Res"],"published-print":{"date-parts":[[2019,9]]},"DOI":"10.1007\/s10100-018-00605-w","type":"journal-article","created":{"date-parts":[[2019,1,8]],"date-time":"2019-01-08T13:36:24Z","timestamp":1546954584000},"page":"615-623","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Distribution of global defensive k-alliances over some graph products"],"prefix":"10.1007","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3315-1759","authenticated-orcid":false,"given":"Mostafa","family":"Tavakoli","sequence":"first","affiliation":[]},{"given":"Sandi","family":"Klav\u017ear","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,1,7]]},"reference":[{"key":"605_CR1","doi-asserted-by":"publisher","first-page":"873","DOI":"10.7151\/dmgt.1952","volume":"37","author":"SE Anderson","year":"2017","unstructured":"Anderson SE, Guo Y, Tenney A, Wash KA (2017) Prime factorization and domination in the hierarchical product of graphs. Discuss Math Graph Theory 37:873\u2013890","journal-title":"Discuss Math Graph Theory"},{"key":"605_CR2","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.disc.2017.06.015","volume":"341","author":"SE Anderson","year":"2018","unstructured":"Anderson SE, Nagpal S, Wash K (2018) Domination in the hierarchical product and Vizing\u2019s conjecture. Discrete Math 341:20\u201324","journal-title":"Discrete Math"},{"key":"605_CR3","first-page":"591","volume":"64","author":"M Arezoomand","year":"2010","unstructured":"Arezoomand M, Taeri B (2010) Applications of generalized hierarchical product of graphs in computing the Szeged index of chemical graphs. MATCH Commun Math Comput Chem 64:591\u2013602","journal-title":"MATCH Commun Math Comput Chem"},{"key":"605_CR4","doi-asserted-by":"publisher","first-page":"3871","DOI":"10.1016\/j.disc.2008.10.028","volume":"309","author":"L Barri\u00e8re","year":"2009","unstructured":"Barri\u00e8re L, Dalf\u00f3 C, Fiol MA, Mitjana M (2009) The generalized hierarchical product of graphs. Discrete Math 309:3871\u20133881","journal-title":"Discrete Math"},{"key":"605_CR5","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719796","volume-title":"Graph classes: a survey","author":"A Brandstadt","year":"1999","unstructured":"Brandstadt A, Le VB, Spinrad JP (1999) Graph classes: a survey. SIAM, Philadelphia"},{"key":"605_CR6","first-page":"225","volume":"107","author":"RG Eballe","year":"2012","unstructured":"Eballe RG, Aldema RM, Paluga EM, Rulete RF, Jamil FP (2012) Global defensive alliances in the join, corona and composition of graphs. Ars Comb. 107:225\u2013245","journal-title":"Ars Comb."},{"key":"605_CR7","doi-asserted-by":"publisher","first-page":"322","DOI":"10.1007\/BF01844162","volume":"4","author":"R Frucht","year":"1970","unstructured":"Frucht R, Harary F (1970) On the coronas of two graphs. Aequationes Math 4:322\u2013324","journal-title":"Aequationes Math"},{"key":"605_CR8","doi-asserted-by":"publisher","DOI":"10.1201\/b10959","volume-title":"Handbook of product graphs","author":"R Hammack","year":"2011","unstructured":"Hammack R, Imrich W, Klav\u017ear S (2011) Handbook of product graphs, 2nd edn. CRC Press, Boca Raton","edition":"2"},{"key":"605_CR9","volume-title":"Fundamentals of domination in graphs","author":"TW Haynes","year":"1998","unstructured":"Haynes TW, Hedetniemi ST, Slater PJ (1998) Fundamentals of domination in graphs. Marcel Dekker, New York"},{"key":"605_CR10","doi-asserted-by":"publisher","first-page":"108","DOI":"10.1186\/1471-2105-7-108","volume":"7","author":"TW Haynes","year":"2006","unstructured":"Haynes TW, Knisley D, Seier E, Zou Y (2006) A quantitative analysis of secondary RNA structure using domination based parameters on trees. BMC Bioinform 7:108","journal-title":"BMC Bioinform"},{"key":"605_CR11","doi-asserted-by":"publisher","first-page":"586","DOI":"10.13001\/1081-3810.1395","volume":"20","author":"Y Hou","year":"2010","unstructured":"Hou Y, Shiu W-C (2010) The spectrum of the edge corona of two graphs. Electron J Linear Algebra 20:586\u2013594","journal-title":"Electron J Linear Algebra"},{"key":"605_CR12","doi-asserted-by":"publisher","first-page":"224","DOI":"10.1016\/0095-8956(86)90079-1","volume":"40","author":"J Kratochv\u00edl","year":"1986","unstructured":"Kratochv\u00edl J (1986) Perfect codes over graphs. J Comb Theory Ser B 40:224\u2013228","journal-title":"J Comb Theory Ser B"},{"key":"605_CR13","first-page":"157","volume":"48","author":"P Kristiansen","year":"2004","unstructured":"Kristiansen P, Hedetniemi SM, Hedetniemi ST (2004) Alliances in graphs. J Comb Math Comb Comput 48:157\u2013177","journal-title":"J Comb Math Comb Comput"},{"key":"605_CR14","doi-asserted-by":"publisher","first-page":"793","DOI":"10.12988\/imf.2016.67102","volume":"11","author":"S Rinurwati","year":"2016","unstructured":"Rinurwati S, Suprajitno H (2016) General results of local metric dimensions of edge-corona of graphs. Int Math Forum 11:793\u2013799","journal-title":"Int Math Forum"},{"key":"605_CR15","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1016\/j.dam.2008.02.006","volume":"157","author":"JA Rodr\u00edguez-Vel\u00e1zquez","year":"2009","unstructured":"Rodr\u00edguez-Vel\u00e1zquez JA, Sigarreta JM (2009) Global defensive \n                    \n                      \n                    \n                    $$k$$\n                    \n                      \n                        k\n                      \n                    \n                  -alliances in graphs. Discrete Appl Math 157:211\u2013218","journal-title":"Discrete Appl Math"},{"key":"605_CR16","first-page":"183","volume":"154","author":"KH Shafique","year":"2002","unstructured":"Shafique KH, Dutton RD (2002) On satisfactory partitioning of graphs. Congr Numer 154:183\u2013194","journal-title":"Congr Numer"},{"key":"605_CR17","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1007\/s11587-013-0162-8","volume":"63","author":"M Tavakoli","year":"2014","unstructured":"Tavakoli M, Rahbarnia F, Ashrafi AR (2014) Applications of the generalized hierarchical product of graphs in computing the vertex and edge PI indices of chemical graphs. Ric Mat 63:59\u201365","journal-title":"Ric Mat"},{"key":"605_CR18","first-page":"43","volume":"3","author":"M Tavakoli","year":"2014","unstructured":"Tavakoli M, Rahbarnia F, Ashrafi AR (2014) Studying the corona product of graphs under some graph invariants. Trans Comb 3:43\u201349","journal-title":"Trans Comb"},{"key":"605_CR19","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/0012-365X(93)E0092-I","volume":"135","author":"Y-N Yeh","year":"1994","unstructured":"Yeh Y-N, Gutman I (1994) On the sum of all distances in composite graphs. Discrete Math 135:359\u2013365","journal-title":"Discrete Math"},{"key":"605_CR20","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1016\/j.dam.2012.08.009","volume":"161","author":"IG Yero","year":"2013","unstructured":"Yero IG, Rodr\u00edguez-Vel\u00e1zquez JA (2013) Computing global offensive alliances in Cartesian product graphs. Discrete Appl Math 161:284\u2013293","journal-title":"Discrete Appl Math"},{"key":"605_CR21","first-page":"141","volume":"105","author":"IG Yero","year":"2017","unstructured":"Yero IG, Rodr\u00edguez-Vel\u00e1zquez JA (2017) A survey on alliances in graphs: defensive alliances. Util Math 105:141\u2013172","journal-title":"Util Math"},{"key":"605_CR22","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1007\/s10114-011-9075-1","volume":"27","author":"IG Yero","year":"2011","unstructured":"Yero IG, Bermudo S, Rodr\u00edguez-Vel\u00e1zquez JA, Sigarreta JM (2011) Partitioning a graph into defensive \n                    \n                      \n                    \n                    $$k$$\n                    \n                      \n                        k\n                      \n                    \n                  -alliances. Acta Math Sin (Engl Ser) 27:73\u201382","journal-title":"Acta Math Sin (Engl Ser)"},{"key":"605_CR23","first-page":"21","volume":"2","author":"IG Yero","year":"2017","unstructured":"Yero IG, Jakovac M, Kuziak D (2017) On global (strong) defensive alliances in some product graphs. Commun Comb Optim 2:21\u201333","journal-title":"Commun Comb Optim"}],"container-title":["Central European Journal of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10100-018-00605-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10100-018-00605-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10100-018-00605-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,2,3]],"date-time":"2020-02-03T09:06:21Z","timestamp":1580720781000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10100-018-00605-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,7]]},"references-count":23,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,9]]}},"alternative-id":["605"],"URL":"https:\/\/doi.org\/10.1007\/s10100-018-00605-w","relation":{},"ISSN":["1435-246X","1613-9178"],"issn-type":[{"type":"print","value":"1435-246X"},{"type":"electronic","value":"1613-9178"}],"subject":[],"published":{"date-parts":[[2019,1,7]]},"assertion":[{"value":"7 January 2019","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}