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Menger\u2019s theorem reveals the relationship between connectivity and disjoint paths in a graph. Disjoint paths not only avoid communication bottlenecks, but also provide alternative paths in case of vertex failures. Let [Formula: see text] [Formula: see text] be a [Formula: see text]-dimensional sub-bubble-sort star of [Formula: see text]. In this paper, we show that [Formula: see text] is strongly Menger (edge) connected. Later, we show that the connectivity and edge-connectivity of [Formula: see text] are uniformly [Formula: see text]. In addition, we show that the 1-extra connectivity of [Formula: see text] is [Formula: see text] for [Formula: see text].<\/jats:p>","DOI":"10.1142\/s0129626425500070","type":"journal-article","created":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T22:15:27Z","timestamp":1748470527000},"source":"Crossref","is-referenced-by-count":0,"title":["Fault-tolerant of Bubble-sort Star Graph Based on Subgraph Fault Pattern"],"prefix":"10.1142","volume":"35","author":[{"given":"Shiying","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan, Shanxi 030031, P.R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoli","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan, Shanxi 030031, P.R. 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