Polynomial time recognition of vertices contained in all (or no) maximum dissociation sets of a tree

In a graph $ G $, a dissociation set is a subset of vertices which induces a subgraph with vertex degree at most 1. Finding a dissociation set of maximum cardinality in a graph is NP-hard even for bipartite graphs and is called the maximum dissociation set problem. The complexity of the maximum diss...

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Published in:AIMS mathematics Vol. 7; no. 1; pp. 569 - 578
Main Authors: Tu, Jianhua, Zhang, Lei, Du, Junfeng, Lang, Rongling
Format: Journal Article
Language:English
Published: AIMS Press 01.01.2022
Subjects:
independent set
maximum dissociation set
polynomial time algorithm
tree
ISSN:2473-6988, 2473-6988
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Abstract In a graph $ G $, a dissociation set is a subset of vertices which induces a subgraph with vertex degree at most 1. Finding a dissociation set of maximum cardinality in a graph is NP-hard even for bipartite graphs and is called the maximum dissociation set problem. The complexity of the maximum dissociation set problem in various sub-classes of graphs has been extensively studied in the literature. In this paper, we study the maximum dissociation problem from different perspectives and characterize the vertices belonging to all maximum dissociation sets, and no maximum dissociation set of a tree. We present a linear time recognition algorithm which can determine whether a given vertex in a tree is contained in all (or no) maximum dissociation sets of the tree. Thus for a tree with $ n $ vertices, we can find all vertices belonging to all (or no) maximum dissociation sets of the tree in $ O(n^2) $ time.
AbstractList In a graph $ G $, a dissociation set is a subset of vertices which induces a subgraph with vertex degree at most 1. Finding a dissociation set of maximum cardinality in a graph is NP-hard even for bipartite graphs and is called the maximum dissociation set problem. The complexity of the maximum dissociation set problem in various sub-classes of graphs has been extensively studied in the literature. In this paper, we study the maximum dissociation problem from different perspectives and characterize the vertices belonging to all maximum dissociation sets, and no maximum dissociation set of a tree. We present a linear time recognition algorithm which can determine whether a given vertex in a tree is contained in all (or no) maximum dissociation sets of the tree. Thus for a tree with $ n $ vertices, we can find all vertices belonging to all (or no) maximum dissociation sets of the tree in $ O(n^2) $ time.
Author Tu, Jianhua
Lang, Rongling
Zhang, Lei
Du, Junfeng
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Cites_doi 10.1007/s13119-011-0002-7
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CorporateAuthor Department of Mathematics, Beijing University of Chemical Technology, Beijing 100029, China
Key Laboratory of Tibetan Information Processing and Machine Translation, Qinghai Province, XiNing 810008, China
Key Laboratory of Tibetan Information Processing, Ministry of Education, XiNing 810008, China
School of Electronics and Information Engineering, Beihang University, Beijing 100191, China
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China
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Snippet In a graph $ G $, a dissociation set is a subset of vertices which induces a subgraph with vertex degree at most 1. Finding a dissociation set of maximum...
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SubjectTerms independent set
maximum dissociation set
polynomial time algorithm
tree
Title Polynomial time recognition of vertices contained in all (or no) maximum dissociation sets of a tree
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