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AIMS Mathematics

2022, Volume 7, Issue 1: 569-578. doi: 10.3934/math.2022036
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Research article

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

  • 1.
    School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China
  • 2.
    Key Laboratory of Tibetan Information Processing and Machine Translation, Qinghai Province, XiNing 810008, China
  • 3.
    Key Laboratory of Tibetan Information Processing, Ministry of Education, XiNing 810008, China
  • 4.
    School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
  • 5.
    Department of Mathematics, Beijing University of Chemical Technology, Beijing 100029, China
  • 6.
    School of Electronics and Information Engineering, Beihang University, Beijing 100191, China
  • Received: 08 August 2021 Accepted: 07 October 2021 Published: 14 October 2021

  • MSC : 05C05, 05C69, 05C85

  • 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.

    Citation: Jianhua Tu, Lei Zhang, Junfeng Du, Rongling Lang. Polynomial time recognition of vertices contained in all (or no) maximum dissociation sets of a tree[J]. AIMS Mathematics, 2022, 7(1): 569-578. doi: 10.3934/math.2022036

    Related Papers:

  • 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.


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