{"id":"https://openalex.org/W3213114993","doi":"https://doi.org/10.3390/sym13112203","title":"Extended Graph of the Fuzzy Topographic Topological Mapping Model","display_name":"Extended Graph of the Fuzzy Topographic Topological Mapping Model","publication_year":2021,"publication_date":"2021-11-18","ids":{"openalex":"https://openalex.org/W3213114993","doi":"https://doi.org/10.3390/sym13112203","mag":"3213114993"},"language":"en","primary_location":{"id":"doi:10.3390/sym13112203","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13112203","pdf_url":"https://www.mdpi.com/2073-8994/13/11/2203/pdf?version=1637327272","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/13/11/2203/pdf?version=1637327272","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023909180","display_name":"Muhammad Zillullah Mukaram","orcid":null},"institutions":[{"id":"https://openalex.org/I4576418","display_name":"University of Technology Malaysia","ror":"https://ror.org/026w31v75","country_code":"MY","type":"education","lineage":["https://openalex.org/I4576418"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Muhammad Zillullah Mukaram","raw_affiliation_strings":["Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia","institution_ids":["https://openalex.org/I4576418"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5060404379","display_name":"Tahir Ahmad","orcid":"https://orcid.org/0000-0003-2675-7952"},"institutions":[{"id":"https://openalex.org/I4576418","display_name":"University of Technology Malaysia","ror":"https://ror.org/026w31v75","country_code":"MY","type":"education","lineage":["https://openalex.org/I4576418"]}],"countries":["MY"],"is_corresponding":true,"raw_author_name":"Tahir Ahmad","raw_affiliation_strings":["Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia"],"raw_orcid":"https://orcid.org/0000-0003-2675-7952","affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia","institution_ids":["https://openalex.org/I4576418"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110582848","display_name":"Norma Alias","orcid":null},"institutions":[{"id":"https://openalex.org/I4576418","display_name":"University of Technology Malaysia","ror":"https://ror.org/026w31v75","country_code":"MY","type":"education","lineage":["https://openalex.org/I4576418"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Norma Alias","raw_affiliation_strings":["Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia","institution_ids":["https://openalex.org/I4576418"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5001169707","display_name":"Noorsufia Abd Shukor","orcid":"https://orcid.org/0000-0002-3228-5413"},"institutions":[{"id":"https://openalex.org/I4576418","display_name":"University of Technology Malaysia","ror":"https://ror.org/026w31v75","country_code":"MY","type":"education","lineage":["https://openalex.org/I4576418"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Noorsufia Abd Shukor","raw_affiliation_strings":["Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia"],"raw_orcid":"https://orcid.org/0000-0002-3228-5413","affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia","institution_ids":["https://openalex.org/I4576418"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5021788465","display_name":"Faridah Mustapha","orcid":null},"institutions":[{"id":"https://openalex.org/I4576418","display_name":"University of Technology Malaysia","ror":"https://ror.org/026w31v75","country_code":"MY","type":"education","lineage":["https://openalex.org/I4576418"]}],"countries":["MY"],"is_corresponding":false,"raw_author_name":"Faridah Mustapha","raw_affiliation_strings":["Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia","institution_ids":["https://openalex.org/I4576418"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5060404379"],"corresponding_institution_ids":["https://openalex.org/I4576418"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.194,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.50943335,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":96},"biblio":{"volume":"13","issue":"11","first_page":"2203","last_page":"2203"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12923","display_name":"Digital Image Processing Techniques","score":0.954800009727478,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12923","display_name":"Digital Image Processing Techniques","score":0.954800009727478,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/conjecture","display_name":"Conjecture","score":0.6606206893920898},{"id":"https://openalex.org/keywords/grid","display_name":"Grid","score":0.6265977621078491},{"id":"https://openalex.org/keywords/extension","display_name":"Extension (predicate logic)","score":0.5371009111404419},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5242698192596436},{"id":"https://openalex.org/keywords/visualization","display_name":"Visualization","score":0.47767502069473267},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.4768620431423187},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.4565344750881195},{"id":"https://openalex.org/keywords/mathematical-induction","display_name":"Mathematical induction","score":0.41728657484054565},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.41319432854652405},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.3937505781650543},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3851657509803772},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.334028959274292},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3220858573913574},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.2757071852684021},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10840508341789246},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.09213832020759583}],"concepts":[{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6606206893920898},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.6265977621078491},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.5371009111404419},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5242698192596436},{"id":"https://openalex.org/C36464697","wikidata":"https://www.wikidata.org/wiki/Q451553","display_name":"Visualization","level":2,"score":0.47767502069473267},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.4768620431423187},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4565344750881195},{"id":"https://openalex.org/C206743973","wikidata":"https://www.wikidata.org/wiki/Q178377","display_name":"Mathematical induction","level":2,"score":0.41728657484054565},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.41319432854652405},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.3937505781650543},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3851657509803772},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.334028959274292},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3220858573913574},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2757071852684021},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10840508341789246},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.09213832020759583},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym13112203","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13112203","pdf_url":"https://www.mdpi.com/2073-8994/13/11/2203/pdf?version=1637327272","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:377aca7a54df40908229f39545c39ef8","is_oa":true,"landing_page_url":"https://doaj.org/article/377aca7a54df40908229f39545c39ef8","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 13, Iss 11, p 2203 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/11/2203/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13112203","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym13112203","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13112203","pdf_url":"https://www.mdpi.com/2073-8994/13/11/2203/pdf?version=1637327272","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7317293862","display_name":null,"funder_award_id":"FRGS/1/2020/STG06/UTM/01/1","funder_id":"https://openalex.org/F4320321709","funder_display_name":"Ministry of Higher Education, Malaysia"}],"funders":[{"id":"https://openalex.org/F4320321709","display_name":"Ministry of Higher Education, Malaysia","ror":"https://ror.org/05mcs2t73"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W3213114993.pdf"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W2924809804","https://openalex.org/W2994109353","https://openalex.org/W3193425473"],"related_works":["https://openalex.org/W2075768550","https://openalex.org/W3022218857","https://openalex.org/W2399035783","https://openalex.org/W2369178846","https://openalex.org/W2370289839","https://openalex.org/W2082716031","https://openalex.org/W3161230432","https://openalex.org/W2933494595","https://openalex.org/W4313347705","https://openalex.org/W1568049691"],"abstract_inverted_index":{"Fuzzy":[0],"topological":[1,16],"topographic":[2],"mapping":[3],"(FTTM)":[4],"is":[5,31,37,75,125,172],"a":[6,12,40,76,133],"mathematical":[7],"model":[8],"which":[9],"consists":[10],"of":[11,14,28,34,46,57,71,79,83,102,113,118,142,159,179],"set":[13],"homeomorphic":[15],"spaces":[17],"designed":[18],"to":[19,92,110,165],"solve":[20],"the":[21,49,55,84,89,100,123,129,143,152,160,167,180],"neuro":[22],"magnetic":[23],"inverse":[24],"problem.":[25],"A":[26,69],"sequence":[27],"FTTM,":[29,47],"FTTMn,":[30],"an":[32],"extension":[33],"FTTM":[35,85,104],"that":[36],"arranged":[38],"in":[39],"symmetrical":[41],"form.":[42],"The":[43,60,154,169],"special":[44,77],"characteristic":[45],"namely":[48],"homeomorphisms":[50],"between":[51],"its":[52],"components,":[53],"allows":[54],"generation":[56],"new":[58],"FTTM.":[59],"generated":[61,103],"FTTMs":[62],"can":[63],"be":[64],"represented":[65],"as":[66],"pseudo":[67,72,105],"graphs.":[68],"graph":[70,80],"degree":[73,106],"zero":[74,107],"type":[78],"where":[81],"each":[82],"components":[86,114],"differs":[87],"from":[88],"one":[90],"adjacent":[91],"it.":[93],"Previous":[94],"researchers":[95],"have":[96],"investigated":[97],"and":[98,115,140,149,157],"conjectured":[99],"number":[101,112,117],"with":[108],"respect":[109],"n":[111],"k":[116],"versions.":[119],"In":[120],"this":[121],"paper,":[122],"conjecture":[124],"proven":[126],"analytically":[127],"for":[128],"first":[130],"time":[131],"using":[132],"newly":[134],"developed":[135,150,155],"grid-based":[136,145,170],"method.":[137],"Some":[138],"definitions":[139,156],"properties":[141,158],"novel":[144],"method":[146,161],"are":[147,162],"introduced":[148],"along":[151],"way.":[153],"then":[163],"assembled":[164],"prove":[166],"conjecture.":[168,181],"technique":[171],"simple":[173],"yet":[174],"offers":[175],"some":[176],"visualization":[177],"features":[178]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2022,"cited_by_count":2}],"updated_date":"2026-05-22T06:13:13.366637","created_date":"2025-10-10T00:00:00"}