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We prove that $g_{1} \\leq g_{2}$ and that there is also a multicurve of cardinality $g_{2} - g_{1}$ contained in every element of the image. We also prove that if $n_{1} = 0$ and $g_{1} = g_{2}$, then the map $\\widetilde{\\varphi}$ obtained by filling the punctures of $S_{2}$, is induced by a homeomorphism of "},"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":"1611.09986","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-11-30T03:30:54Z","cross_cats_sorted":[],"title_canon_sha256":"90d93229a65f4e5d27ffdc14a3da730924193c62f0fcca2d397d47168ebf50dd","abstract_canon_sha256":"a01939d9e42ea69be5fa666c91f8e8ee03867d8581d0ae3cd5a14bb8472a7388"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:47.756355Z","signature_b64":"CmhKAJHXg4mjaSoKgx/ehwTnlu3y+SydeWKXFduSdThRDFZVIEqQ+tuHd4MaEe8yIPoSUstnhAz6S4xHqPIGCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b5f982a66404213c38fe0d281f8889faf620dd12117af2b931ed9a8ede22b13","last_reissued_at":"2026-05-18T00:36:47.755843Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:47.755843Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Alternating maps on Hatcher-Thurston graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jes\\'us Hern\\'andez Hern\\'andez","submitted_at":"2016-11-30T03:30:54Z","abstract_excerpt":"Let $S_{1}$ and $S_{2}$ be connected orientable surfaces of genus $g_{1}, g_{2} \\geq 3$, $n_{1},n_{2} \\geq 0$ punctures, and empty boundary. Let also $\\varphi: \\mathcal{HT}(S_{1}) \\rightarrow \\mathcal{HT}(S_{2})$ be an edge-preserving alternating map between their Hatcher-Thurston graphs. We prove that $g_{1} \\leq g_{2}$ and that there is also a multicurve of cardinality $g_{2} - g_{1}$ contained in every element of the image. 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