{"scopus_import":"1","volume":347,"date_published":"2024-03-19T00:00:00Z","quality_controlled":"1","article_number":"113962","citation":{"ista":"Campbell R, Hörsch F, Moore B. 2024. Decompositions into two linear forests of bounded lengths. Discrete Mathematics. 347(6), 113962.","mla":"Campbell, Rutger, et al. “Decompositions into Two Linear Forests of Bounded Lengths.” <i>Discrete Mathematics</i>, vol. 347, no. 6, 113962, Elsevier, 2024, doi:<a href=\"https://doi.org/10.1016/j.disc.2024.113962\">10.1016/j.disc.2024.113962</a>.","short":"R. Campbell, F. Hörsch, B. Moore, Discrete Mathematics 347 (2024).","chicago":"Campbell, Rutger, Florian Hörsch, and Benjamin Moore. “Decompositions into Two Linear Forests of Bounded Lengths.” <i>Discrete Mathematics</i>. Elsevier, 2024. <a href=\"https://doi.org/10.1016/j.disc.2024.113962\">https://doi.org/10.1016/j.disc.2024.113962</a>.","ama":"Campbell R, Hörsch F, Moore B. Decompositions into two linear forests of bounded lengths. <i>Discrete Mathematics</i>. 2024;347(6). doi:<a href=\"https://doi.org/10.1016/j.disc.2024.113962\">10.1016/j.disc.2024.113962</a>","apa":"Campbell, R., Hörsch, F., &#38; Moore, B. (2024). Decompositions into two linear forests of bounded lengths. <i>Discrete Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.disc.2024.113962\">https://doi.org/10.1016/j.disc.2024.113962</a>","ieee":"R. Campbell, F. Hörsch, and B. Moore, “Decompositions into two linear forests of bounded lengths,” <i>Discrete Mathematics</i>, vol. 347, no. 6. Elsevier, 2024."},"title":"Decompositions into two linear forests of bounded lengths","intvolume":"       347","article_processing_charge":"No","publication_identifier":{"issn":["0012-365X"]},"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2301.11615","open_access":"1"}],"author":[{"first_name":"Rutger","last_name":"Campbell","full_name":"Campbell, Rutger"},{"last_name":"Hörsch","first_name":"Florian","full_name":"Hörsch, Florian"},{"full_name":"Moore, Benjamin","id":"6dc1a1be-bf1c-11ed-8d2b-d044840f49d6","first_name":"Benjamin","last_name":"Moore"}],"day":"19","language":[{"iso":"eng"}],"abstract":[{"text":"For some k∈Z≥0∪{∞}, we call a linear forest k-bounded if each of its components has at most k edges. We will say a (k,ℓ)-bounded linear forest decomposition of a graph G is a partition of E(G) into the edge sets of two linear forests Fk,Fℓ where Fk is k-bounded and Fℓ is ℓ-bounded. We show that the problem of deciding whether a given graph has such a decomposition is NP-complete if both k and ℓ are at least 2, NP-complete if k≥9 and ℓ=1, and is in P for (k,ℓ)=(2,1). Before this, the only known NP-complete cases were the (2,2) and (3,3) cases. Our hardness result answers a question of Bermond et al. from 1984. We also show that planar graphs of girth at least nine decompose into a linear forest and a matching, which in particular is stronger than 3-edge-colouring such graphs.","lang":"eng"}],"oa":1,"year":"2024","_id":"15163","status":"public","article_type":"original","department":[{"_id":"MaKw"}],"oa_version":"Preprint","doi":"10.1016/j.disc.2024.113962","type":"journal_article","date_updated":"2024-03-25T08:09:43Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"We wish to thank Dániel Marx and András Sebő for making us aware of the results in [8] and some clarifications on them.","issue":"6","publication":"Discrete Mathematics","publisher":"Elsevier","external_id":{"arxiv":["2301.11615"]},"month":"03","date_created":"2024-03-24T23:00:58Z","publication_status":"epub_ahead"}