{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:22:56Z","timestamp":1760059376606,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,10]],"date-time":"2025-06-10T00:00:00Z","timestamp":1749513600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Qinghai Province","award":["QHKLYC-GDCXCY-2022-249","2023-QGY-6"],"award-info":[{"award-number":["QHKLYC-GDCXCY-2022-249","2023-QGY-6"]}]},{"name":"Qinghai University Science Foundation of China","award":["QHKLYC-GDCXCY-2022-249","2023-QGY-6"],"award-info":[{"award-number":["QHKLYC-GDCXCY-2022-249","2023-QGY-6"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>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 q. 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 m is in the range of [5,4n\u221210] and [n2\u2212n+4,n2]. At the same time, the locally optimal structures in the range of the edge number m with (4n\u221210,n2\u2212n+4) 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.<\/jats:p>","DOI":"10.3390\/axioms14060457","type":"journal-article","created":{"date-parts":[[2025,6,10]],"date-time":"2025-06-10T06:48:56Z","timestamp":1749538136000},"page":"457","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Judgment Criteria for Reliability Comparison of Three-Terminal Graphs with High Edge Failure Probability"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2267-7069","authenticated-orcid":false,"given":"Sun","family":"Xie","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, Qinghai University, Xining 810016, China"}]},{"given":"Haixing","family":"Zhao","sequence":"additional","affiliation":[{"name":"The Computer College, Qinghai MinZu University, Xining 810007, China"}]},{"given":"Jun","family":"Yin","sequence":"additional","affiliation":[{"name":"The College of Computer, Qinghai Normal University, Xining 810016, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8551-3339","authenticated-orcid":false,"given":"Jinyu","family":"Zou","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Qinghai University, Xining 810016, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1002\/net.22004","article-title":"Roots of two-terminal reliability polynomials","volume":"78","author":"Brown","year":"2021","journal-title":"Networks"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1016\/j.dam.2024.06.021","article-title":"On the construction of locally most reliable two-terminal graphs","volume":"356","author":"Gong","year":"2024","journal-title":"Discret. 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