[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81960-en":3,"doc-seo-81960-105":31,"detail-sidebar-cat-0-en-105":93},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81960,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","The Minimum Dominating Set Problem on Bipartite Circle Graphs: Complexity and Approximation","A circle graph models chord intersections on a circle, and a dominating set D covers every vertex outside D by adjacency. Minimum dominating set is NP-hard on circle graphs, and this work studies the same optimization on bipartite circle graphs defined by chord representations using two non-intersecting color classes. NP-hardness is proved via reduction from Planar Monotone 3-SAT, alongside a polynomial-time 2-approximation and a local-search PTAS.","arXiv :2607 .0625 1v 1 [ cs .CG] 7 Jul 2026  \nThe Minimum Dominating Set Problem on Bipartite Circle Graphs: Complexity and Approximation  \nA. Karim Abu-Affash∗ Paz Carmi† Joseph S. B. Mitchell‡  \nJuly 8, 2026  \nAbstract  \nA circle graph is the intersection graph of a set of chords in a circle. A dominating set of a graph G = (V, E) is a subset D ⊆ V such that every vertex in V \\ D is adjacent to at least one vertex of D. Computing a minimum dominating set is known to be NP-hard on circle graphs.  \nIn this paper, we study the minimum dominating set problem on bipartite circle graphs, namely, circle graphs admitting a chord representation in which the chords can be partitioned into two color classes such that no two chords of the same color intersect. We prove that the problem remains NP-hard for this restricted graph class by a reduction from Planar Monotone 3-SAT. On the positive side, we present a polynomial-time 2-approximation algorithm and develop a polynomial-time approximation scheme (PTAS) based on local search.  \n1 Introduction  \nGiven a graph G = (V, E), a subset D of V is a dominating set of G if every vertex in V \\ D is adjacent to at least one vertex of D. The minimum dominating set (MDS) problem is to compute a dominating set D of minimum cardinality. The MDS problem is fundamental in the field of graph theory. The problem has real-world applications in network design, facility location, and the analysis of social networks [6,8,19,23,29] . It has been shown that the MDS problem is NP-hard on general graphs [15], as are many other variants of domination, including total dominating set, connected dominating set, independent dominating set, and dominating clique [5,15,18] . All of these variants, except for independent dominating set, are also NP-hard on chordal graphs [1,3,13,24,25] . For a summary of the complexity of these variants on other classes of graphs, see Table 1 in [7] .  \nA circle graph G = (V, E) is the intersection graph of a set C of chords in a circle, such that each vertex v ∈ V uniquely corresponds to a chord, and there is an edge (u, v) ∈ E if and only if the two chords corresponding to u and v intersect. C is called a chord intersection model of the circle graph G. Equivalently, an interval model of G is a set I of intervals located on a horizontal line L, such that the vertices of V uniquely correspond to the intervals of I, and two vertices in V are adjacent if and only if the corresponding intervals intersect, i.e., overlap, but neither contains the  \n∗ Software Engineering Department, Shamoon College of Engineering, Beer-Sheva, Israel, [abuaa1@sce.ac.il](abuaa1@sce.ac.il).  \n†The Stein Faculty of Computer and Information Science, Ben-Gurion University, Beer-Sheva, Israel, [carmip@bgu.ac.il](carmip@bgu.ac.il).  \n‡Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY, USA, [joseph.mitchell@stonybrook.edu](joseph.mitchell@stonybrook.edu).  \nother. The three models of a circle graph (as a graph, as a set of chords, and as a set of intervals) are equivalent via linear time transformations [16] . Thus, w.l.o.g., when specifying instances of problems, we assume the availability of the most convenient model.  \nCircle graphs have been extensively studied in the literature [2,10,11,16,21,27] . Many problems that are NP-hard in general graphs become solvable in polynomial time when restricted to circle graphs, including maximum clique and maximum independent set [16], minimum feedback vertex set [17], and dominating clique [21] . On the other hand, some problems remain NP-hard in circle graphs, such as Hamiltonian cycle [9], minimum clique cover [22], and k-coloring for k ≥ 4 [28] .  \nDetermining the complexity of the MDS problem in circle graphs was first asked by Johnson [20] in 1985 . This question remained open until Keil [21] proved in 1993 that the MDS problem is NPcomplete. Later, Damian and Pemmaraju [11] provided a (2 + ε)-approximation scheme for the proble","cbCaibJ4VuUbNCEK","https://ap.wps.com/l/cbCaibJ4VuUbNCEK","pdf",799719,7,1,20,"English","en",105,"# Abstract\n# Introduction\n# NP-hardness\n# Approximation","[{\"question\":\"What are circle graphs and how is domination defined on them?\",\"answer\":\"A circle graph represents intersections among chords on a circle: vertices correspond to chords, and edges connect chord pairs that intersect. A dominating set D ensures every vertex not in D is adjacent to at least one vertex in D.\"},{\"question\":\"How does the paper restrict the problem to bipartite circle graphs?\",\"answer\":\"It considers circle graphs admitting a chord representation where chords are split into two colors (red and blue) with no intersections within the same color class, yielding bipartite structure. The interval model is also used with red and blue intervals partitioned accordingly.\"},{\"question\":\"What complexity and approximation results are proved for the minimum dominating set problem?\",\"answer\":\"The paper proves NP-hardness persists for bipartite circle graphs using a reduction from Planar Monotone 3-SAT. It also provides a polynomial-time 2-approximation algorithm and develops a polynomial-time approximation scheme (PTAS) based on local search.\"}]","The Minimum Dominating Set Problem on Bipartite Circle Graphs: Complexity and Approximation | PDF",1784177297,50,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"the-minimum-dominating-set-problem-on-bipartite-circle-graphs-complexity-and-approximation","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/the-minimum-dominating-set-problem-on-bipartite-circle-graphs-complexity-and-approximation/81960/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What are circle graphs and how is domination defined on them?","Question",{"text":77,"@type":78},"A circle graph represents intersections among chords on a circle: vertices correspond to chords, and edges connect chord pairs that intersect. A dominating set D ensures every vertex not in D is adjacent to at least one vertex in D.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How does the paper restrict the problem to bipartite circle graphs?",{"text":82,"@type":78},"It considers circle graphs admitting a chord representation where chords are split into two colors (red and blue) with no intersections within the same color class, yielding bipartite structure. The interval model is also used with red and blue intervals partitioned accordingly.",{"name":84,"@type":75,"acceptedAnswer":85},"What complexity and approximation results are proved for the minimum dominating set problem?",{"text":86,"@type":78},"The paper proves NP-hardness persists for bipartite circle graphs using a reduction from Planar Monotone 3-SAT. It also provides a polynomial-time 2-approximation algorithm and develops a polynomial-time approximation scheme (PTAS) based on local search.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,112,116,120,123,127,130,134],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":108,"doc_module":4,"doc_module_name":47,"category_name":109,"show_sort_weight":110,"slug":111},5,"Comic",60,"comic",{"id":113,"doc_module":4,"doc_module_name":47,"category_name":114,"show_sort_weight":30,"slug":115},6,"Technology","technology",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":117,"show_sort_weight":118,"slug":119},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":47,"category_name":125,"show_sort_weight":22,"slug":126},9,"Religion & Spirituality","religion-spirituality",{"id":22,"doc_module":4,"doc_module_name":47,"category_name":128,"show_sort_weight":22,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":47,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":47,"category_name":136,"show_sort_weight":108,"slug":137},19,"General","general"]