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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图分解为匹配线性森林和有界线性森林的复杂性

确定图是否可以边分解为匹配和k<； 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></mrow></ mrow></mrow></ mrow></mrow></ mrow></math>；-有界线性森林最近由Campbell， Hörsch，对于每k≥9，多项式时间（NP）是完全的<；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0002”威利:位置= "方程/ jgt23208 -数学- 0002. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> k< / mi> & lt; mo> \ unicode {x02265} & lt; / mo> & lt; mn> 9 & lt; / mn> & lt; / mrow> & lt; / mrow> & lt; / math>，且k = 1时在多项式时间内可解，2< math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23208:jgt23208- jgt23208-math-0003.png"><mrow>< jgt23208- jgt23208-math-0003.png"><mrow>< jgt23208- jgt23208- jgt23208-math-0003.png"><mrow>< jgt23208- jgt23208- jgt23208- jgt23208-math-0003.png"><mrow>< /mrow>< mrow> =</ mrow>< m>1</ m>< m>,</ mrow><；。在本文的第一部分，我们通过证明这个问题对于每个k≥3是np完备的来缩小这个差距<；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23208:jgt23208-math-0004”威利:位置= "方程/ jgt23208 -数学- 0004. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> k< / mi> & lt; mo> \ unicode {x02265} & lt; / mo> & lt; mn> 3 & lt; / mn> & lt; / mrow> & lt; / mrow> & lt; / math>。在论文的第二部分，我们证明了决定一个图是否可以边分解为一个匹配和一个k<； 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>< /mrow></mrow></ mrow></mrow></ mrow></mrow></math>；-有界星林对任意k∈N∪{∞}多项式可解<math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23208:jgt23208- jgt23208-math-0006.png“><mrow><mrow>< mrow>< / mrow>< / mrow>< / mrow>< / mrow><mrow><mi mathvariant=”双划”>N</ m02208}</mo><mrow><mo class="MathClass-open">{</ mo0222a}</mo><mrow>< / mrow><类= " MathClass-close "祝辞}& lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>，回答Campbell、Hörsch和Moore在同一篇论文中提出的另一个问题。

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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 .