The Complexity of Decomposing a Graph into a Matching and a Bounded Linear Forest

IF 0.9 3区数学 Q2 MATHEMATICS Journal of Graph Theory Pub Date : 2025-01-08 DOI:10.1002/jgt.23208

Agnijo Banerjee, João Pedro Marciano, Adva Mond, Jan Petr, Julien Portier

{"title":"The Complexity of Decomposing a Graph into a Matching and a Bounded Linear Forest","authors":"Agnijo Banerjee, João Pedro Marciano, Adva Mond, Jan Petr, Julien Portier","doi":"10.1002/jgt.23208","DOIUrl":null,"url":null,"abstract":"<div>\n \n Deciding whether a graph can be edge-decomposed into a matching and a <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0001\" wiley:location=\"equation/jgt23208-math-0001.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math>-bounded linear forest was recently shown by Campbell, Hörsch, and Moore to be nonedeterministic Polynomial time (NP)-complete for every <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>9</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0002\" wiley:location=\"equation/jgt23208-math-0002.png\"><mrow><mrow><mi>k</mi><mo>\\unicode{x02265}</mo><mn>9</mn></mrow></mrow></math></annotation>\n </semantics></math>, and solvable in polynomial time for <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>=</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n \n <mn>2</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0003\" wiley:location=\"equation/jgt23208-math-0003.png\"><mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mrow></math></annotation>\n </semantics></math>. In the first part of this paper, we close this gap by showing that this problem is NP-complete for every <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>3</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0004\" wiley:location=\"equation/jgt23208-math-0004.png\"><mrow><mrow><mi>k</mi><mo>\\unicode{x02265}</mo><mn>3</mn></mrow></mrow></math></annotation>\n </semantics></math>. In the second part of the paper, we show that deciding whether a graph can be edge-decomposed into a matching and a <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0005\" wiley:location=\"equation/jgt23208-math-0005.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math>-bounded star forest is polynomially solvable for any <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>∈</mo>\n \n <mi>N</mi>\n \n <mo>∪</mo>\n \n <mrow>\n <mo>{</mo>\n \n <mi>∞</mi>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0006\" wiley:location=\"equation/jgt23208-math-0006.png\"><mrow><mrow><mi>k</mi><mo>\\unicode{x02208}</mo><mi mathvariant=\"double-struck\">N</mi><mo>\\unicode{x0222A}</mo><mrow><mo class=\"MathClass-open\">{</mo><mi>\\unicode{x0221E}</mi><mo class=\"MathClass-close\">}</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math>, answering another question by Campbell, Hörsch, and Moore from the same paper.\n </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"76-87"},"PeriodicalIF":0.9000,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23208","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Deciding whether a graph can be edge-decomposed into a matching and a $k$ -bounded linear forest was recently shown by Campbell, Hörsch, and Moore to be nonedeterministic Polynomial time (NP)-complete for every $k \geq 9$ , and solvable in polynomial time for $k = 1, 2$ . In the first part of this paper, we close this gap by showing that this problem is NP-complete for every $k \geq 3$ . In the second part of the paper, we show that deciding whether a graph can be edge-decomposed into a matching and a $k$ -bounded star forest is polynomially solvable for any $k \in N \cup {\infty}$ , answering another question by Campbell, Hörsch, and Moore from the same paper.

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .

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