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<p>A graph <i>G</i> is minimally <i>t</i>-tough if the toughness of <i>G</i> is <i>t</i> and the deletion of any edge from <i>G</i> decreases its toughness, where <i>t</i> is a positive real number. It has been conjectured that there exists a vertex of degree 2 in every minimally 1-tough graph. In this paper, we completely determine the structure of minimally 1-tough graphs with independence number not exceeding 3 and thus confirm the conjecture for this class of graphs.</p>

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The Structure of Minimally 1-Tough Graphs with Small Independence Number

  • 期刊论文

  • 发表日期：

    2025年2月18日

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  • Shiyu Cao,
  • Jing Chen,
  • Wei Zheng

摘要

A graph G is minimally t-tough if the toughness of G is t and the deletion of any edge from G decreases its toughness, where t is a positive real number. It has been conjectured that there exists a vertex of degree 2 in every minimally 1-tough graph. In this paper, we completely determine the structure of minimally 1-tough graphs with independence number not exceeding 3 and thus confirm the conjecture for this class of graphs.