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Let $G = (V,E)$ be a vertex transitive graph, let $A \\subseteq V$ be a finite vertex-set with $|A| \\le |V|/2$ and $|\\{v \\in V \\setminus A : {$u \\sim v$ for some $u \\in A$} \\}|\\le k$. We show that whenever the diameter of $G$ is at least $31(k+1)^2$, either $|A| \\le 2k^3+k^2$, or $G$ has a ring-like structure (with bounded parameters), and $A$ is efficiently contained in an interval. This theorem may be viewed as a rough characterization, generalizing a"},"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":"1110.4885","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-21T19:54:13Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"e6bb1c141abcbcc844e7bfc9a4b0a5bf814ba7837ad99ac68d899f562077fd53","abstract_canon_sha256":"3c91c3d820443724499fe31ced394e7f5d7e13b5e8679ab2b4375a547e5c7250"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:34.803162Z","signature_b64":"PDl9gRj8cjB/VS/aJ/ssk/YWuNiPZ8nRBh2LpMCxoUINiXNdmza1vpwCPee59khS6YDON1KpkmrzmskbCWqRBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a6e2c01f6726c310755ae2d82da5f60eab5464b3c17fc529ad96f31a7260ee0","last_reissued_at":"2026-05-18T04:10:34.802452Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:34.802452Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Small separations in vertex transitive graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Bojan Mohar, Matt DeVos","submitted_at":"2011-10-21T19:54:13Z","abstract_excerpt":"Let $k$ be an integer. We prove a rough structure theorem for separations of order at most $k$ in finite and infinite vertex transitive graphs. Let $G = (V,E)$ be a vertex transitive graph, let $A \\subseteq V$ be a finite vertex-set with $|A| \\le |V|/2$ and $|\\{v \\in V \\setminus A : {$u \\sim v$ for some $u \\in A$} \\}|\\le k$. We show that whenever the diameter of $G$ is at least $31(k+1)^2$, either $|A| \\le 2k^3+k^2$, or $G$ has a ring-like structure (with bounded parameters), and $A$ is efficiently contained in an interval. 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