[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81757-en":3,"doc-seo-81757-105":30,"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":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":13,"seo_description":14,"update_tm":28,"read_time":29},81757,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Determining the Complexity of Chromatic Sum in Classes Defined by a Set of Forbidden Graphs","Chromatic Sum asks whether a graph admits a vertex colouring whose colour-sum across all vertices is at most a given threshold. The work investigates the computational complexity of this decision problem on graph classes characterized by forbidding specified substructures under various containment relations. It fully classifies complexity for H-minor-free, H-topological-minor-free, and H-subgraph-free settings via three meta-classifications, and proves new NP-completeness for suitable planar-subcubic subdivisions.","arXiv :2607 .00263v1 [math .CO] 30 Jun 2026  \nDetermining the Complexity of Chromatic Sum in Classes Defined by a Set of Forbidden Graphs  \nClément Dallard \\#   \nDepartment of Informatics, University of Fribourg, Fribourg, Switzerland Daniël Paulusma \\#   \nDepartment of Computer Science, Durham University, Durham, United Kingdom Erik Jan van Leeuwen \\#   \nDepartment of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands  \n~~ Abstract ~~  \nThe Chromatic Sum problem asks, given a graph G and an integer k, whether G admits a colouring c with sum Pv∈V c (v) ≤ k. We study the complexity of Chromatic Sum on graph classes defined by some set of forbidden graphs. First, we show that three known frameworks fully classify the complexity of Chromatic Sum on H-minor-free graphs and H-topological-minor-free graphs for any set of graphs H, and on H-subgraph-free graphs for any finite set of graphs H. To show this, we prove a new NP-completeness result for Chromatic Sum on certain subdivisions of planar subcubic graphs. Next, we consider other containment relations. We formalise a novel framework of problems that are NP-complete for planar graphs as well as for graphs of bounded independence number. For every problem in this framework, we obtain an almost complete complexity classification on H-induced-minor-free graphs, H-induced-topological-minor-free graphs, and H-free graphs for every graph H. We show that Chromatic Sum belongs to this framework, as do several other problems. We also define a more fine-grained framework for the induced subgraph relation. We apply this to obtain a complete complexity classification for Chromatic Sum on H-free graphs, as well as for several other problems. We justify the choice of this framework by proving that Chromatic Sum is NP-complete for graphs of clique-width at most 3. This result complements a known polynomial-time result for graphs of clique-width at most 2.  \n2012 ACM Subject Classification Mathematics of computing → Graph theory; Theory of computa  \ntion → Graph algorithms analysis; Theory of computation → Problems, reductions and completeness Keywords and phrases complexity dichotomy, graph colouring, induced subgraph, subgraph Funding Daniël Paulusma: Supported by the Leverhulme Trust Grant RPG-2024-182 .  \nAcknowledgements The authors wish to thank Hans Bodlaender and Danny Hermelin for their helpful ideas towards the proofs of Theorem 10 and 24 .  \n 1  Introduction  \nLet G = (V, E) be a graph. A function c : V → {1, 2 , . . .} is a colouring of G if c(u)  c (v) for every edge uv ∈ E. The sum of c is defined as Pv∈V c (v) . The chromatic sum of G is the minimum sum over all colourings of G. The Chromatic Sum (or Sum Colouring) problem, formulated as a decision problem, asks whether, given a graph G and integer k , the graph G admits a colouring with sum at most k. The Chromatic Sum problem is motivated by applications in VLSI design [56, 63, 64] and scheduling [6, 33] .  \nThe study of Chromatic Sum in the past decades has yielded various insights into its complexity, see e.g. [6,7,9,12,13,21,25,32,35,36,43,44,50–52,56,61–64] . Notably, Chromatic Sum is NP-complete [44], even on planar graphs [32, 50] (even if they are bipartite and of maximum degree 5 [51], and even if they are cubic [50]), interval graphs [52, 64], split graphs [61, 62], penny graphs, unit square graphs, and unit disk graphs [13] . Yet, it is  \n2 Chromatic Sum in Classes Defined by a Set of Forbidden Graphs  \npolynomial-time solvable on trees [44], graphs of bounded treewidth [36], proper interval graphs [56], P4-free graphs [36], and several generalizations of P4-free graphs [12, 62] .  \nWe consider the complexity of Chromatic Sum for graph classes defined by a set of forbidden graphs. The way these graphs are not contained in the input graph is determined by a containment relation. Four prominent graph operations, vertex deletion (VD), edge deletion (ED), edge contraction (EC), and vertex diss","cbCaisC8VZBtO47Q","https://ap.wps.com/l/cbCaisC8VZBtO47Q","pdf",830200,3,1,20,"English","en",105,"# Introduction\n## Problem definition and motivation\n## Containment relations and forbidden graph frameworks\n## Prior complexity results and research gap","[{\"question\":\"What does the Chromatic Sum problem ask?\",\"answer\":\"Given a graph G and integer k, it asks whether there exists a colouring of the vertices such that the sum of assigned colours over all vertices is at most k.\"},{\"question\":\"How does the paper study Chromatic Sum for forbidden graph classes?\",\"answer\":\"It analyzes how complexity changes when graphs are restricted to avoid certain structures, formalized through containment relations derived from operations like vertex/edge deletion, contraction, and vertex dissolution.\"},{\"question\":\"What is the main high-level contribution to complexity classification?\",\"answer\":\"The paper shows a full classification for Chromatic Sum on several major families of forbidden-graph classes using known meta-classifications, while introducing new NP-completeness results and developing finer-grained frameworks for other containment relations.\"}]",1784175851,50,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"determining-the-complexity-of-chromatic-sum-in-classes-defined-by-a-set-of-forbidden-graphs","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/determining-the-complexity-of-chromatic-sum-in-classes-defined-by-a-set-of-forbidden-graphs/81757/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","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 Chromatic Sum problem ask?","Question",{"text":75,"@type":76},"Given a graph G and integer k, it asks whether there exists a colouring of the vertices such that the sum of assigned colours over all vertices is at most k.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper study Chromatic Sum for forbidden graph classes?",{"text":80,"@type":76},"It analyzes how complexity changes when graphs are restricted to avoid certain structures, formalized through containment relations derived from operations like vertex/edge deletion, contraction, and vertex dissolution.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main high-level contribution to complexity classification?",{"text":84,"@type":76},"The paper shows a full classification for Chromatic Sum on several major families of forbidden-graph classes using known meta-classifications, while introducing new NP-completeness results and developing finer-grained frameworks for other containment 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