Judgment Criteria for Reliability Comparison of Three-Terminal Graphs with High Edge Failure Probability
A three-terminal graph is defined as a simple graph comprising three specified target vertices. The reliability of three-terminal graphs represents the probability that these three target vertices remain connected, given that each edge fails independently with a constant probability <i>q</i...
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2025-06-01
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| author | Sun Xie Haixing Zhao Jun Yin Jinyu Zou |
| author_facet | Sun Xie Haixing Zhao Jun Yin Jinyu Zou |
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| description | A three-terminal graph is defined as a simple graph comprising three specified target vertices. The reliability of three-terminal graphs represents the probability that these three target vertices remain connected, given that each edge fails independently with a constant probability <i>q</i>. In this paper, we focus on exploring the characteristics of more reliable three-terminal graphs when the edge failure probability approaches 1. Three reliability comparison criteria are proposed to characterize the locally most reliable three-terminal graph progressively when the number of edges <i>m</i> is in the range of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>5</mn><mo>,</mo><mn>4</mn><mi>n</mi><mo>−</mo><mn>10</mn><mo>]</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac linethickness="0pt"><mi>n</mi><mn>2</mn></mfrac></mstyle></mfenced><mo>−</mo><mi>n</mi><mo>+</mo><mn>4</mn><mo>,</mo><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac linethickness="0pt"><mi>n</mi><mn>2</mn></mfrac></mstyle></mfenced><mo>]</mo></mrow></semantics></math></inline-formula>. At the same time, the locally optimal structures in the range of the edge number <i>m</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>4</mn><mi>n</mi><mo>−</mo><mn>10</mn><mo>,</mo><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac linethickness="0pt"><mi>n</mi><mn>2</mn></mfrac></mstyle></mfenced><mo>−</mo><mi>n</mi><mo>+</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> are restricted to six specific classes of graphs. Furthermore, based on these criteria, a method is introduced to search local optimal structures and offer a theoretical foundation for constructing optimal networks and repairing faulty ones. |
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| spelling | doaj-art-cbd2ee586bd04414813f94328f98659b2025-08-20T03:27:26ZengMDPI AGAxioms2075-16802025-06-0114645710.3390/axioms14060457Judgment Criteria for Reliability Comparison of Three-Terminal Graphs with High Edge Failure ProbabilitySun Xie0Haixing Zhao1Jun Yin2Jinyu Zou3School of Mathematics and Physics, Qinghai University, Xining 810016, ChinaThe Computer College, Qinghai MinZu University, Xining 810007, ChinaThe College of Computer, Qinghai Normal University, Xining 810016, ChinaSchool of Mathematics and Physics, Qinghai University, Xining 810016, ChinaA three-terminal graph is defined as a simple graph comprising three specified target vertices. The reliability of three-terminal graphs represents the probability that these three target vertices remain connected, given that each edge fails independently with a constant probability <i>q</i>. In this paper, we focus on exploring the characteristics of more reliable three-terminal graphs when the edge failure probability approaches 1. Three reliability comparison criteria are proposed to characterize the locally most reliable three-terminal graph progressively when the number of edges <i>m</i> is in the range of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>5</mn><mo>,</mo><mn>4</mn><mi>n</mi><mo>−</mo><mn>10</mn><mo>]</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac linethickness="0pt"><mi>n</mi><mn>2</mn></mfrac></mstyle></mfenced><mo>−</mo><mi>n</mi><mo>+</mo><mn>4</mn><mo>,</mo><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac linethickness="0pt"><mi>n</mi><mn>2</mn></mfrac></mstyle></mfenced><mo>]</mo></mrow></semantics></math></inline-formula>. At the same time, the locally optimal structures in the range of the edge number <i>m</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>4</mn><mi>n</mi><mo>−</mo><mn>10</mn><mo>,</mo><mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true"><mfrac linethickness="0pt"><mi>n</mi><mn>2</mn></mfrac></mstyle></mfenced><mo>−</mo><mi>n</mi><mo>+</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> are restricted to six specific classes of graphs. Furthermore, based on these criteria, a method is introduced to search local optimal structures and offer a theoretical foundation for constructing optimal networks and repairing faulty ones.https://www.mdpi.com/2075-1680/14/6/457three-terminal graphlocally more reliablejudgment criterialocally most reliable graph |
| spellingShingle | Sun Xie Haixing Zhao Jun Yin Jinyu Zou Judgment Criteria for Reliability Comparison of Three-Terminal Graphs with High Edge Failure Probability Axioms three-terminal graph locally more reliable judgment criteria locally most reliable graph |
| title | Judgment Criteria for Reliability Comparison of Three-Terminal Graphs with High Edge Failure Probability |
| title_full | Judgment Criteria for Reliability Comparison of Three-Terminal Graphs with High Edge Failure Probability |
| title_fullStr | Judgment Criteria for Reliability Comparison of Three-Terminal Graphs with High Edge Failure Probability |
| title_full_unstemmed | Judgment Criteria for Reliability Comparison of Three-Terminal Graphs with High Edge Failure Probability |
| title_short | Judgment Criteria for Reliability Comparison of Three-Terminal Graphs with High Edge Failure Probability |
| title_sort | judgment criteria for reliability comparison of three terminal graphs with high edge failure probability |
| topic | three-terminal graph locally more reliable judgment criteria locally most reliable graph |
| url | https://www.mdpi.com/2075-1680/14/6/457 |
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