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Let $\\Delta$ be the maximum degree of $G$. We show that there exists a function $f(\\Delta) = (\\Delta+1)^{\\Delta^2+1}$, so that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint and pairwise anticomplete paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of at most $k \\cdot f(\\De"},"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":"2309.08169","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-09-15T05:34:53Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"c10ad83fd09e86f6842c490a9e36200222dc430216fb233d59a53cfe900ca452","abstract_canon_sha256":"106e3551df7832f173ce9a247dc9f746fe8e17d037da1246e1d38d7d5cd56125"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:51:02.878088Z","signature_b64":"450Zi3ZMz4bgsy45u24Q1cvBA+1zAvuxmgf1b3ydNYtF9hHPNuCho+4NcXyoWlCzvRiwOGMf0WQrXGFpMEDYAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c1e4969efa9e9f13f70a8cb6bf9d36f4f46ff14c956dbd49da0593078851037c","last_reissued_at":"2026-07-05T06:51:02.877688Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:51:02.877688Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Induced Versions of Menger's Theorem on Sparse Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"math.CO","authors_text":"Daniel Lokshtanov, Peter Gartland, Tuukka Korhonen","submitted_at":"2023-09-15T05:34:53Z","abstract_excerpt":"Let $A$ and $B$ be sets of vertices in a graph $G$. Menger's theorem states that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of at most $k-1$ vertices. Let $\\Delta$ be the maximum degree of $G$. 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