[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83777-en":3,"doc-seo-83777-105":30,"detail-sidebar-cat-0-en-105":92},{"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},83777,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Sequential Cable Constructions and Linear Rank-Width","Sequential cable constructions and linear rank-width are studied through a split-free cable language where live cables enforce uniform GF(2) row behavior across each active cut. A width-w play yields a birth-order layout whose cutrank is bounded by floor(w/2), implying the sequential split-free width is at least twice the linear rank-width. For the first nontrivial level, connected graphs have linear rank-width ≤ 1 exactly when they admit a stream, equivalently a singleton-birth play of width ≤ 4.","arXiv :2607 .04141v1 [math .CO] 5 Jul 2026  \nDiscrete Mathematics and Theoretical Computer Science DMTCS, submission version  \nSequential cable constructions and linear rank-width Antonios Kalampakas  \nAmerican University of the Middle East, Kuwait  \nAbstract. We introduce split-free cable terms and cable plays, a sequential graph-construction language whose live cables impose uniform GF(2)-row behaviour across the current cut. Every play of width w gives a birth-order layout whose cutrank is at most half of w, rounded down, so the sequential split-free width is at least twice the linear rank-width. At the first nontrivial level we prove an exact characterization: a connected graph with at least two vertices has linear rankwidth at most one exactly when it admits a stream, equivalently a singleton-birth play of width at most four. We show that unrestricted term width and sequential width differ unboundedly on trees, calibrate the construction on the net graph, and formulate an affine upper-bound conjecture relating sequential split-free width to linear rank-width. For the rank-two case we prove a two-accumulator scheduling criterion that yields width-six plays under a natural future-uniformity hypothesis.  \nKeywords: linear rank-width, rank-width, graph constructions, graph expressions, vertex-minors, cutrank  \n1 Introduction  \nRank-width, introduced by Oum and Seymour [16, 19], measures a graph by the GF(2) ranks of the cuts of a branch-decomposition. Linear rank-width restricts the decomposition to a linear order of the vertices, taking the maximum cutrank over prefixes. Equivalently, it is the caterpillar or path-like version of rank-width. Thus rank-width and linear rank-width stand to one another as treewidth and path-width do.  \nRank-width has become one of the main ways to make dense graph structure usable. Its first role is algorithmic: rank-width and clique-width are tied by the inequalities  \nrw(G) ≤ cw(G) ≤ 2rw(G)+1 − 1 ,  \nso bounded rank-width is equivalent to bounded clique-width [19, 18] . Since clique-width expressions support algorithmic meta-theorems for MSO 1-definable graph properties and related optimization problems on graph classes of bounded clique-width [7], rank-width gives a route to those algorithms through cutrank decompositions. This is strengthened by decomposition algorithms: for fixed width there are algorithms that either produce a rank-decomposition of bounded width or certify that the width is too large [19, 17], and exact fixed-parameter algorithms for branch- and rankdecompositions were developed by Hliněný and Oum [11] . Thus rank-width is not merely descriptive, it supplies certificates on which dynamic programming and clique-width methods can operate.  \nIts second role is structural. Rank-width is adapted to the vertex-minor and pivot-minor orders in much the same way that tree-width is adapted to graph minors. Oum proved that, for every fixed k, the class of graphs of rank-width at most k has a finite set of excluded vertex-minors [16] . The corresponding linear theory is also robust: linear rank-width has finite vertex-minor obstruction sets for each fixed level [12], although these sets become very large in higher levels [13] . Kwon and Oum further showed that every graph of rank-width at most k is a pivot-minor of a graph of tree-width at most 2k, and every graph of linear rank-width at most k is a pivot-minor of a graph of path-width at most k+1 . In particular, graphs of linear rank-width at most one are precisely the vertex-minors of paths [15] . These results explain why linear rank-width is a natural rank-theoretic analogue of path-width rather than a merely ad hoc linearization.  \nAt the first nontrivial level the linear hierarchy is well understood from several viewpoints. Ganian’s thread graphs describe the graphs of linear rank-width at most one [8] and Adler, Farley,  \nSequential cable constructions and linear rank-width 2  \nFigure 1: The net graph used as a small calibrat","cbCaieFYzpSYISEL","https://ap.wps.com/l/cbCaieFYzpSYISEL","pdf",393150,6,1,17,"English","en",105,"# Introduction\n## Rank-width and linear rank-width\n## Structural and obstruction-theoretic background\n## Known characterizations at low levels\n## Motivation for a sequential cable language","[{\"question\":\"What is the main contribution of the split-free cable terms and cable plays?\",\"answer\":\"They define a sequential graph-construction language where live cables impose uniform GF(2)-row behavior across the current cut, enabling direct control of cutrank through play width.\"},{\"question\":\"How does play width relate to linear rank-width?\",\"answer\":\"A play of width w yields a birth-order layout whose cutrank is at most floor(w/2), so the sequential split-free width is at least twice the linear rank-width.\"},{\"question\":\"What is proven for the first nontrivial level of linear rank-width?\",\"answer\":\"A connected graph with at least two vertices has linear rank-width at most one exactly when it admits a stream, equivalently when it has a singleton-birth play of width at most 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is the main contribution of the split-free cable terms and cable plays?","Question",{"text":76,"@type":77},"They define a sequential graph-construction language where live cables impose uniform GF(2)-row behavior across the current cut, enabling direct control of cutrank through play width.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does play width relate to linear rank-width?",{"text":81,"@type":77},"A play of width w yields a birth-order layout whose cutrank is at most floor(w/2), so the sequential split-free width is at least twice the linear rank-width.",{"name":83,"@type":74,"acceptedAnswer":84},"What is proven for the first nontrivial level of linear rank-width?",{"text":85,"@type":77},"A connected graph with at least two vertices has linear rank-width at most one exactly when it admits a stream, equivalently when it has a singleton-birth play of width at most 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