{"id":"https://openalex.org/W2523500016","doi":"https://doi.org/10.1051/ita/2016015","title":"The average lower reinforcement number of a graph","display_name":"The average lower reinforcement number of a graph","publication_year":2016,"publication_date":"2016-04-01","ids":{"openalex":"https://openalex.org/W2523500016","doi":"https://doi.org/10.1051/ita/2016015","mag":"2523500016"},"language":"en","primary_location":{"id":"doi:10.1051/ita/2016015","is_oa":false,"landing_page_url":"https://doi.org/10.1051/ita/2016015","pdf_url":null,"source":{"id":"https://openalex.org/S6941720","display_name":"RAIRO - Theoretical Informatics and Applications","issn_l":"0988-3754","issn":["0988-3754","1290-385X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319748","host_organization_name":"EDP Sciences","host_organization_lineage":["https://openalex.org/P4310319748"],"host_organization_lineage_names":["EDP Sciences"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"RAIRO - Theoretical Informatics and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://www.numdam.org/item/10.1051/ita/2016015.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5065654666","display_name":"Tufan Turac\u0131","orcid":"https://orcid.org/0000-0002-6159-0935"},"institutions":[{"id":"https://openalex.org/I173761726","display_name":"Karab\u00fck University","ror":"https://ror.org/04wy7gp54","country_code":"TR","type":"education","lineage":["https://openalex.org/I173761726"]}],"countries":["TR"],"is_corresponding":true,"raw_author_name":"Tufan Turaci","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Karab\u00fck University, 78050, Karab\u00fck, Turkey"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Karab\u00fck University, 78050, Karab\u00fck, Turkey","institution_ids":["https://openalex.org/I173761726"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5056662953","display_name":"Ersin Aslan","orcid":"https://orcid.org/0000-0002-5496-2075"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ersin Aslan","raw_affiliation_strings":["Turgutlu Vocational Training School, Celal Bayar University, 45400 Manisa, Turkey"],"affiliations":[{"raw_affiliation_string":"Turgutlu Vocational Training School, Celal Bayar University, 45400 Manisa, Turkey","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5065654666"],"corresponding_institution_ids":["https://openalex.org/I173761726"],"apc_list":null,"apc_paid":null,"fwci":1.713,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.83628223,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"50","issue":"2","first_page":"135","last_page":"144"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9890000224113464,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9890000224113464,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9787999987602234,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9423999786376953,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7087254524230957},{"id":"https://openalex.org/keywords/reinforcement","display_name":"Reinforcement","score":0.6538171172142029},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5756206512451172},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5230824947357178},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4656667709350586},{"id":"https://openalex.org/keywords/complement-graph","display_name":"Complement graph","score":0.41525799036026},{"id":"https://openalex.org/keywords/simple-graph","display_name":"Simple graph","score":0.4147696793079376},{"id":"https://openalex.org/keywords/graph-power","display_name":"Graph power","score":0.3852221965789795},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.20183610916137695}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7087254524230957},{"id":"https://openalex.org/C67203356","wikidata":"https://www.wikidata.org/wiki/Q1321905","display_name":"Reinforcement","level":2,"score":0.6538171172142029},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5756206512451172},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5230824947357178},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4656667709350586},{"id":"https://openalex.org/C168291704","wikidata":"https://www.wikidata.org/wiki/Q902252","display_name":"Complement graph","level":5,"score":0.41525799036026},{"id":"https://openalex.org/C2993105083","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Simple graph","level":3,"score":0.4147696793079376},{"id":"https://openalex.org/C149530733","wikidata":"https://www.wikidata.org/wiki/Q5597091","display_name":"Graph power","level":4,"score":0.3852221965789795},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.20183610916137695},{"id":"https://openalex.org/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","level":1,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1051/ita/2016015","is_oa":false,"landing_page_url":"https://doi.org/10.1051/ita/2016015","pdf_url":null,"source":{"id":"https://openalex.org/S6941720","display_name":"RAIRO - Theoretical Informatics and Applications","issn_l":"0988-3754","issn":["0988-3754","1290-385X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319748","host_organization_name":"EDP Sciences","host_organization_lineage":["https://openalex.org/P4310319748"],"host_organization_lineage_names":["EDP Sciences"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"RAIRO - Theoretical Informatics and Applications","raw_type":"journal-article"},{"id":"pmh:oai:numdam.org:ITA_2016__50_2_135_0","is_oa":true,"landing_page_url":"http://www.numdam.org/articles/10.1051/ita/2016015/","pdf_url":"https://www.numdam.org/item/10.1051/ita/2016015.pdf","source":{"id":"https://openalex.org/S4306402264","display_name":"French digital mathematics library (Numdam)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I70900168","host_organization_name":"Universit\u00e9 Savoie Mont Blanc","host_organization_lineage":["https://openalex.org/I70900168"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Text"}],"best_oa_location":{"id":"pmh:oai:numdam.org:ITA_2016__50_2_135_0","is_oa":true,"landing_page_url":"http://www.numdam.org/articles/10.1051/ita/2016015/","pdf_url":"https://www.numdam.org/item/10.1051/ita/2016015.pdf","source":{"id":"https://openalex.org/S4306402264","display_name":"French digital mathematics library (Numdam)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I70900168","host_organization_name":"Universit\u00e9 Savoie Mont Blanc","host_organization_lineage":["https://openalex.org/I70900168"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Text"},"sustainable_development_goals":[{"display_name":"Peace, Justice and strong institutions","score":0.5899999737739563,"id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2523500016.pdf"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W186137167","https://openalex.org/W859034116","https://openalex.org/W1964392667","https://openalex.org/W1970112701","https://openalex.org/W1975099798","https://openalex.org/W1976227376","https://openalex.org/W2012572972","https://openalex.org/W2017030366","https://openalex.org/W2047797996","https://openalex.org/W2048460015","https://openalex.org/W2059273818","https://openalex.org/W2071934684","https://openalex.org/W2091150736","https://openalex.org/W2127379330","https://openalex.org/W2155628394","https://openalex.org/W2167562659","https://openalex.org/W2183973290","https://openalex.org/W2345472562","https://openalex.org/W2352152999","https://openalex.org/W2406639167","https://openalex.org/W2528215271","https://openalex.org/W3113315233"],"related_works":["https://openalex.org/W3003396734","https://openalex.org/W2024593554","https://openalex.org/W2436733907","https://openalex.org/W1526657483","https://openalex.org/W2371887249","https://openalex.org/W2751086833","https://openalex.org/W1591685386","https://openalex.org/W2170810426","https://openalex.org/W2025517193","https://openalex.org/W2072333043"],"abstract_inverted_index":{"Let":[0],"G":[1,52,70],"=":[2],"(V(G),E(G))":[3],"be":[4],"a":[5,13,16,20,25,90],"simple":[6],"undirected":[7],"graph.":[8,21],"The":[9,37,64],"reinforcement":[10,32,39,49,67,87],"number":[11,33,68,88],"of":[12,19,34,48,58,69,89],"graph":[14,61,91,101],"is":[15,44,71],"vulnerability":[17],"parameter":[18],"We":[22],"have":[23],"investigated":[24],"refinement":[26],"that":[27,53],"involves":[28],"the":[29,45,55,59,84,95],"average":[30,65,85],"lower":[31,38,66,86],"this":[35,80],"parameter.":[36],"number,":[40],"denoted":[41],"by":[42,73],"re\u2217(G),":[43],"minimum":[46],"cardinality":[47],"set":[50],"in":[51],"contains":[54],"edge":[56],"e\u2217":[57],"complement":[60],"G\u0305":[62],".":[63],"defined":[72],"rav(G)=1/|E(G\u0305)|":[74],"\u2211e**":[75],"\u2208":[76],"E(G\u0305)":[77],"re*(G)":[78],".In":[79],"paper,":[81],"we":[82,93],"define":[83],"and":[92],"present":[94],"exact":[96],"values":[97],"for":[98],"some":[99],"well\u2212known":[100],"families.":[102]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}