[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82349-en":3,"doc-seo-82349-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82349,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Closing the Complexity Gap for Exact Domatic Number at Three and Four","The exact domatic-number problem determines whether a given graph G satisfies dom(G)=k. Prior work established DP-completeness for every fixed k≥5, leaving k=3 and k=4 open. The paper closes this gap via a polynomial-time reduction from 3SAT that produces graphs with domatic number 4 when satisfiable and 2 when unsatisfiable, never yielding dom(G)=3. Constructive gadget-based proofs encode assignments and clause satisfaction, and soundness shows any sufficiently large domatic partition enforces consistency, yielding DP-completeness of Exact-3-DNP and Exact-4-DNP and, for all fixed k≥3, completes the classification.","arXiv :2607 .09442v 1 [ cs .CC] 10 Jul 2026  \nClosing the Complexity Gap for Exact Domatic Number at Three and Four  \nHolger Spakowski  \nDepartment of Mathematics and Applied Mathematics University of Cape Town  \nRondebosch 7701, South Africa  \n[Holger. Spakowski@uct. ac. za](Holger. Spakowski@uct. ac. za)  \nAbstract  \nThe exact domatic-number problem asks, for a fixed integer k , whether a given graph G satisfies dom(G) = k. Riege and Rothe proved DP-completeness for every fixed k ≥ 5, while the cases k = 3 and k = 4 remained open. We close this classification gap. The main ingredient is a polynomial-time reduction from 3SAT whose output graphs have domatic number 4 in the satisfiable case and domatic number 2 in the unsatisfiable case; in particular, the reduction never produces a graph of domatic number 3. This directly realizes the route suggested by Riege and Rothe for closing the remaining cases. Together with a simpler three-versus-two reduction, this yields DP-completeness of Exact-3-DNP and Exact-4-DNP. The proofs are constructive and give explicit graph gadgets whose local domination constraints encode truth assignments and clause satisfaction. The soundness arguments show conversely that any sufficiently large domatic partition enforces the intended consistency conditions and therefore yields a satisfying assignment. Consequently, Exact-k-DNP is DP-complete for every fixed k ≥ 3, completing the fixed-value classification from k = 3 onward.  \nKeywords: domatic number; exact optimization problems; DP-completeness; boolean hierarchy; graph domination; polynomial-time reductions  \n1 Introduction  \nA dominating set in a graph G is a set of vertices that reaches every vertex of G within distance at most one. The domatic number problem asks for a partition of the vertex set into as many disjoint dominating sets as  \npossible. The maximum number of parts in such a partition is the domatic number of G, denoted here by dom (G) . The problem was already discussed in the early work of Cockayne and Hedetniemi on domination in graphs [CH75, CH77], and it has since become a standard graph partitioning problem. Its interpretation is particularly natural in network settings: a dominating set may model a set of facilities or transmitting stations that can serve all vertices of the network, and a domatic partition then corresponds to several disjoint such service layers [CH77, RR06b] .  \nFor fixed k, the decision problem k-DNP asks whether dom (G) ≥ k. It is known that k-DNP is NP-complete for every fixed k ≥ 3, whereas 2-DNPis polynomial-time decidable [GJ79, KS94] . In this paper we study the corresponding exact problems. For fixed k, let  \nExact-k-DNP = {G : dom(G) = k} .  \nExact versions of NP-optimization problems often naturally lie not merely in NP, but in the class DP, the second level of the boolean hierarchy over NP: one has to certify that the optimum is at least a given value and also that it is not larger than that value. This viewpoint goes back to the work of Papadimitriou and Yannakakis on DP, to the subsequent development of the boolean hierarchy by Cai et al., and to Wagner’s technique for proving hardness in the levels of the boolean hierarchy [PY84, CGH+88 , CGH+89 , Wag87] .  \nRiege and Rothe initiated the systematic study of exact domatic-number problems within this framework. They proved that Exact-i-DNP is DPcomplete for every fixed i ≥ 5, while Exact-2-DNP is coNP-complete [RR06b] . Moreover, Exact-1-DNP is polynomial-time decidable, since for every nonempty graph dom (G) = 1 if and only if G has an isolated vertex. Hence the only fixed exact values not covered by the previous classification were  \ni = 3 and i = 4 ,  \nwhich were left open in the original work of Riege and Rothe [RR06b] . The same gap is also recorded in later expositions [RR06a, Rot05] . This gap is analogous in spirit to the earlier gap in exact graph colorability: Wagner proved DP-completeness for sufficiently large exact chromatic numbers, ","cbCaidp9pbNhyuFp","https://ap.wps.com/l/cbCaidp9pbNhyuFp","pdf",374423,1,26,"English","en",105,"# Introduction\n## Related Work","[{\"question\":\"What does the exact domatic-number problem ask for?\",\"answer\":\"For a fixed integer k, it asks whether a given graph G satisfies dom(G)=k.\"},{\"question\":\"How are the open cases k=3 and k=4 resolved?\",\"answer\":\"By constructing polynomial-time reductions from 3SAT whose outputs have domatic number 4 in the satisfiable case and 2 in the unsatisfiable case, while never producing domatic number 3.\"},{\"question\":\"Why does the avoidance of domatic number 3 matter?\",\"answer\":\"It directly eliminates the remaining classification gap highlighted in earlier work and enables proofs that Exact-3-DNP and Exact-4-DNP are DP-complete.\"}]",1784179808,66,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"closing-the-complexity-gap-for-exact-domatic-number-at-three-and-four","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/closing-the-complexity-gap-for-exact-domatic-number-at-three-and-four/82349/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the exact domatic-number problem ask for?","Question",{"text":75,"@type":76},"For a fixed integer k, it asks whether a given graph G satisfies dom(G)=k.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the open cases k=3 and k=4 resolved?",{"text":80,"@type":76},"By constructing polynomial-time reductions from 3SAT whose outputs have domatic number 4 in the satisfiable case and 2 in the unsatisfiable case, while never producing domatic number 3.",{"name":82,"@type":73,"acceptedAnswer":83},"Why does the avoidance of domatic number 3 matter?",{"text":84,"@type":76},"It directly eliminates the remaining classification gap highlighted in earlier work and enables proofs that Exact-3-DNP and Exact-4-DNP are DP-complete.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]