[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84508-en":3,"doc-seo-84508-105":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},84508,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Exact Algorithms for Edge Deletion to Cactus Graphs and Weighted Variants","Exact exponential-time algorithms are presented for transforming a connected graph into a connected cactus via edge deletion. The unweighted task deletes the minimum number of edges so the remaining spanning subgraph is a connected cactus, where each edge belongs to at most one simple cycle. The work improves a prior O*(3^n)-time and O*(3^n)-space approach to O*(2^n), and extends to weighted deletions under bounded distinct costs, yielding O*(2^n · nO(q)) time/space.","arXiv :2606 . 17578v2 [ cs .DS] 11 Jul 2026  \nExact Algorithms for Edge Deletion to Cactus Graphs and Weighted Variants  \nWenhao Song  \nDepartment of Computer Science, Technion—– Israel Institute of Technology [wenhao.song@campus.technion.ac.il](wenhao.song@campus.technion.ac.il)  \nAbstract. We study exact exponential-time algorithms for Edge Deletion to Cactus. Given a connected graph G, the task is to delete a minimum number of edges so that the remaining spanning graph is a connected cactus. Akhtar and Philip (IWOCA 2026) gave an O∗ (3n )-time algorithm for the unweighted problem, where n is the number of vertices in the input graph and the O ∗ () notation hides polynomial factors.  \nWe improve this algorithm to O ∗ (2n ) time and space. More generally, if the deletion costs take at most q distinct nonnegative real values, then the weighted problem can be solved in O∗ (2n nO (q)) time and space. Thus every fixed number of distinct costs, and in particular the unweighted case, admits a faster exact algorithm. For nonnegative integer costs of total weight W, we obtain an O ∗ (2n (W + 1)) pseudo-polynomial algorithm, while arbitrary nonnegative real costs admit an O ∗ (3n ) exact algorithm.  \n1 Introduction  \nGraph modification problems ask for a minimum number of local edits that transform an input graph into a graph belonging to a prescribed class. In the edge deletion version, the vertex set is fixed and one seeks a largest spanning subgraph in the target class. Such problems go back at least to Yannakakis [16] and El-Mallah and Colbourn [5], see also Cai [4], Natanzon, Shamir, and Sharan [13], and Fomin and Kratsch [6] .  \nWe study the target class of cactus graphs. A connected graph is a cactus if every edge lies in at most one simple cycle. Equivalently, every block is either a single edge or a simple cycle. Thus cactus graphs sit immediately beyond trees: they allow local cycles while retaining a tree-like global structure through their cut-vertex decomposition. They were studied classically under the name Husimi trees [9] .  \nThis paper studies the following problem.  \nEdge Deletion to Cactus  \nInput: A connected simple undirected graph G = (V, E) .  \nTask: Find a minimum cardinality set F ⊆ E such that G − F is a connected cactus.  \nSince only edges are deleted, every feasible solution is spanning. Hence Edge Deletion to Cactus is equivalent to the problem of finding a connected spanning cactus subgraph of G with the maximum possible number of edges.  \nWe also study the weighted version.  \nWeighted Edge Deletion to Cactus  \nInput: A connected simple undirected graph G = (V, E) and a nonnegative  \ndeletion cost function c : E → R≥0 .  \nTask: Find a set F ⊆ E minimizing  \nc (F) =X ce  \ne∈F  \nsuch that G − F is a connected cactus.  \nEquivalently, Weighted Edge Deletion to Cactus asks for a connected spanning cactus H ⊆ G that maximizes c(E(H)) = Pe∈E(H) ce. The unweighted problem is the special case in which every deletion cost is one.  \n1.1 Prior work  \nThe problem is NP-hard on general graphs, as shown by El-Mallah and Colbourn [5] . More recently, Koch, Pardal, and dos Santos studied edge deletion to several tree-like graph classes and proved, among other results, that deletion to cactus graphs remains hard even on bipartite input graphs. They also gave positive results for chordal and quasi-threshold input graphs [12] . These results place cactus deletion problem at a natural boundary: deletion to trees is elementary, while deletion to the slightly larger class of cactus graphs is already computationally hard.  \nThe exact-exponential algorithm of the problem was studied recently by Akhtar and Philip [1] . They considered Edge Deletion to Cactus together with the related Spanning Tree to Cactus problem. They gave a polynomialtime algorithm for the latter using edge intersection graphs of paths in a tree [7,8], which also yields an O ∗ (nn−2) exact algorithm for Edge Deletion to Cactus by enumerating spanning trees. Th","cbCaimeltR4smAdV","https://ap.wps.com/l/cbCaimeltR4smAdV","pdf",370603,1,15,"English","en",105,"# Introduction\n## Problem definition: Edge Deletion to Cactus\n## Problem definition: Weighted Edge Deletion to Cactus\n## Prior work\n## Our results\n## Main theorems","[{\"question\":\"What does “Edge Deletion to Cactus” ask for?\",\"answer\":\"Given a connected undirected graph, it seeks a minimum-cardinality edge set F whose removal leaves a connected cactus spanning subgraph.\"},{\"question\":\"How is the weighted version defined?\",\"answer\":\"Each edge has a nonnegative deletion cost, and the goal is to remove edges with minimum total cost while keeping the remaining graph a connected cactus.\"},{\"question\":\"What runtime improvement is achieved over the previous unweighted algorithm?\",\"answer\":\"The prior O*(3^n)-time algorithm is improved to O*(2^n) time and space for the unweighted case, using an optimized exact-exponential strategy.\"}]",1784196206,38,{"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},"exact-algorithms-for-edge-deletion-to-cactus-graphs-and-weighted-variants","",{"@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/exact-algorithms-for-edge-deletion-to-cactus-graphs-and-weighted-variants/84508/",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 “Edge Deletion to Cactus” ask for?","Question",{"text":75,"@type":76},"Given a connected undirected graph, it seeks a minimum-cardinality edge set F whose removal leaves a connected cactus spanning subgraph.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the weighted version defined?",{"text":80,"@type":76},"Each edge has a nonnegative deletion cost, and the goal is to remove edges with minimum total cost while keeping the remaining graph a connected cactus.",{"name":82,"@type":73,"acceptedAnswer":83},"What runtime improvement is achieved over the previous unweighted algorithm?",{"text":84,"@type":76},"The prior O*(3^n)-time algorithm is improved to O*(2^n) time and space for the unweighted case, using an optimized exact-exponential strategy.","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"]