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We say that a removable edge $e$ in a brick $G$ is $b$-invariant if $b(G-e)=b(G)=1$, where $b(H)$ denotes the number of bricks in the tight cut decomposition of a matching covered graph $H$. An edge of a graph is solitary if it lies in precisely one perfect matching.\n  Lucchesi and Murty proposed the problem of characterizing bricks, distinct from $K_4$, $\\overline{C_6}$ and the Petersen g"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2608.12832","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-08-13T05:10:49Z","cross_cats_sorted":[],"title_canon_sha256":"90d4460d55adb7bd712a94c15baf20a4b3ebd9cd1ef2ed795c56ab477ec54af1","abstract_canon_sha256":"32cc06ad8f8944b77ed7575850c14446aaab1de9e2413c9e260960b84216b42d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-14T00:48:54.914076Z","signature_b64":"EU4mcdeOU8L+x0lf8xDGo6lLEvtiKAJ2IJ5nvRv8aXvB5Qv8O1tUJ0BEgwxTHbzzH/skDNMOdE9/7yim76COCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b9df21fa20e1dc901b6a005699276dc3f9db274e5c5f0dbf78ed8961a2da10c","last_reissued_at":"2026-08-14T00:48:54.911021Z","signature_status":"signed_v1","first_computed_at":"2026-08-14T00:48:54.911021Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bricks that every removable edge is solitary","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fuliang Lu, Jinxin Xue, Jun Ge, Yaxian Zhang","submitted_at":"2026-08-13T05:10:49Z","abstract_excerpt":"A brick is a 3-connected graph $G$ such that $G-u-v$ has a perfect matching for any two distinct vertices $u,v\\in V(G)$. 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