{"id":"https://openalex.org/W7133311733","doi":"https://doi.org/10.48550/arxiv.2603.00692","title":"Parameterized complexity of $r$-Hop, $r$-Step, and $r$-Hop Roman Domination","display_name":"Parameterized complexity of $r$-Hop, $r$-Step, and $r$-Hop Roman Domination","publication_year":2026,"publication_date":"2026-02-28","ids":{"openalex":"https://openalex.org/W7133311733","doi":"https://doi.org/10.48550/arxiv.2603.00692"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2603.00692","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.00692","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2603.00692","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5127997825","display_name":"Sandip Das","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Das, Sandip","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5126690434","display_name":"Sweta Das","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Das, Sweta","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5038722857","display_name":"Sk Samim Islam","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Islam, Sk Samim","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9607999920845032,"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"}},"topics":[{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9607999920845032,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.02419999986886978,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.006599999964237213,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/dominating-set","display_name":"Dominating set","score":0.7185999751091003},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.7103999853134155},{"id":"https://openalex.org/keywords/bipartite-graph","display_name":"Bipartite graph","score":0.5557000041007996},{"id":"https://openalex.org/keywords/parameterized-complexity","display_name":"Parameterized complexity","score":0.47760000824928284},{"id":"https://openalex.org/keywords/vertex-cover","display_name":"Vertex cover","score":0.4569999873638153},{"id":"https://openalex.org/keywords/domination-analysis","display_name":"Domination analysis","score":0.37599998712539673},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.366100013256073},{"id":"https://openalex.org/keywords/existential-quantification","display_name":"Existential quantification","score":0.33500000834465027}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7466999888420105},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7314000129699707},{"id":"https://openalex.org/C146661039","wikidata":"https://www.wikidata.org/wiki/Q2915204","display_name":"Dominating set","level":4,"score":0.7185999751091003},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.7103999853134155},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.5557000041007996},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.47760000824928284},{"id":"https://openalex.org/C40687702","wikidata":"https://www.wikidata.org/wiki/Q11515519","display_name":"Vertex cover","level":3,"score":0.4569999873638153},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4196000099182129},{"id":"https://openalex.org/C83833204","wikidata":"https://www.wikidata.org/wiki/Q5290322","display_name":"Domination analysis","level":4,"score":0.37599998712539673},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.366100013256073},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.33500000834465027},{"id":"https://openalex.org/C67810366","wikidata":"https://www.wikidata.org/wiki/Q835942","display_name":"Interval graph","level":5,"score":0.3077999949455261},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.3043999969959259},{"id":"https://openalex.org/C161677786","wikidata":"https://www.wikidata.org/wiki/Q2478475","display_name":"Neighbourhood (mathematics)","level":2,"score":0.30250000953674316},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.2797999978065491},{"id":"https://openalex.org/C18359143","wikidata":"https://www.wikidata.org/wiki/Q7888149","display_name":"Maximal independent set","level":5,"score":0.2754000127315521},{"id":"https://openalex.org/C2778945305","wikidata":"https://www.wikidata.org/wiki/Q6865502","display_name":"Minimum weight","level":2,"score":0.27320000529289246},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.27079999446868896},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.26739999651908875}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2603.00692","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.00692","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2603.00692","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.00692","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"The":[0,139,149],"\\textsc{Dominating":[1],"Set}":[2],"problem":[3,155,178],"is":[4,60,71,83,112,143,194,224],"a":[5,45,113,128],"classical":[6],"and":[7,14,38,47,207,233,244,267],"extensively":[8],"studied":[9],"topic":[10],"in":[11,68,259],"graph":[12,46],"theory":[13],"theoretical":[15],"computer":[16],"science.":[17],"In":[18],"this":[19],"paper,":[20],"we":[21,214],"examine":[22],"the":[23,174,201,248],"algorithmic":[24],"complexity":[25],"of":[26,31,79,91,103,141,168,188,252],"several":[27],"well-known":[28],"exact-distance":[29],"variants":[30],"domination,":[32],"namely":[33],"\\textsc{$r$-Step":[34,153,203,231],"Domination},":[35,37,204,206],"\\textsc{$r$-Hop":[36,39,150,175,205,208,221,234],"Roman":[40,107,176,185,209,222],"Domination}.":[41],"Let":[42],"$G$":[43,92,111,158,181],"be":[44,52],"let":[48],"$r":[49],"\\geq":[50],"2$":[51],"an":[53,61,84,160,183,256],"integer.":[54],"A":[55],"set":[56,163],"$S":[57],"\\subseteq":[58],"V(G)$":[59],"\\emph{$r$-hop":[62,106],"dominating":[63,86,108,162,166,186],"set}":[64,87],"if":[65,88],"every":[66,89,121,198,228],"vertex":[67,78,90,102,122,129],"$V(G)\\setminus":[69],"S$":[70],"at":[72,94,99,131,170,190],"distance":[73,95,132],"exactly":[74,96,133],"$r$":[75,97,134],"from":[76,98,135],"some":[77],"$S$.":[80,104],"Similarly,":[81],"$S$":[82],"\\emph{$r$-step":[85],"lies":[93],"least":[100],"one":[101],"An":[105],"function}":[109],"on":[110,262],"function":[114,187],"$f":[115],"\\colon":[116],"V(G)\\to\\{0,1,2\\}$":[117],"such":[118],"that":[119,196,216],"for":[120,197,217,227],"$v$":[123,136],"with":[124,137,264],"$f(v)=0$,":[125],"there":[126],"exists":[127],"$u$":[130],"$f(u)=2$.":[138],"\\emph{weight}":[140],"$f$":[142],"defined":[144],"as":[145],"$f(V)=\\sum_{v\\in":[146],"V(G)}":[147],"f(v)$.":[148],"Domination}":[151,177,210,223,232,235],"(respectively,":[152,164],"Domination})":[154],"asks":[156,179],"whether":[157,180],"admits":[159,182,255],"$r$-hop":[161,184],"$r$-step":[165],"set)":[167],"size":[169],"most":[171,191],"$k$,":[172],"while":[173],"weight":[189],"$k$.":[192],"It":[193],"known":[195],"$r\\ge":[199,219,229],"2$,":[200,220,230],"problems":[202,254],"are":[211],"\\textsc{NP}-complete.":[212],"First":[213],"prove":[215],"all":[218],"\\textsc{W[2]}-complete.":[225],"Furthermore,":[226],"remain":[236],"\\textsc{W[2]}-hard":[237],"even":[238],"when":[239],"restricted":[240],"to":[241],"bipartite":[242],"graphs":[243,263],"chordal":[245],"graphs.":[246],"Unless":[247],"ETH":[249],"fails,":[250],"none":[251],"these":[253],"algorithm":[257],"running":[258],"time":[260],"$2^{o(n+m)}$":[261],"$n$":[265],"vertices":[266],"$m$":[268],"edges.":[269]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-03-04T00:00:00"}