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The minimum cardinality of an $x$-geodominating set of $G$ is defined as the $x$-geodomination number of $G$, $g_x(G)$, and an $x$-geodominating set of cardinality $g_x(G)$ is called a $g_x$-set and it is known that it is unique for each vertex $x$. We prove that, in any graph $G$, the $g_x$-set associated to a vertex $x$ is the set of boundary vertices of $x$, that is $\\partial(x)= \\{v \\in V(G) : \\forall "},"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":"1311.3804","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-15T10:51:06Z","cross_cats_sorted":[],"title_canon_sha256":"12290ae5ec3cc3ae253b9cf97f061ee14b0c23bb2da2e9f1f4cb39cbbba22cc3","abstract_canon_sha256":"b93738793d21e4e28f76d6159373996b854b95a677911c91688be919346ec995"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:02.986461Z","signature_b64":"kKG8a5nLhaGiliIRoNd+6DCp9GLzkJv3A5VfnIDJCIw8bfxtU3bdaWOQt5wM/21rUzyM12fQrYMDPvX8FTnmAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c35009685328d70a52f630d1ce669812591c58bf6986b5dbbc9d39b50358ef8e","last_reissued_at":"2026-05-18T03:07:02.986068Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:02.986068Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the boundary as an $x$-geodominating set in graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J. C\\'aceres, M.L. Puertas, M. Morales","submitted_at":"2013-11-15T10:51:06Z","abstract_excerpt":"Given a graph $G$ and a vertex $x\\in V(G)$, a vertex set $S \\subseteq V(G)$ is an $x$-geodominating set of $G$ if each vertex $v\\in V(G)$ lies on an $x-y$ geodesic for some element $y\\in S$. The minimum cardinality of an $x$-geodominating set of $G$ is defined as the $x$-geodomination number of $G$, $g_x(G)$, and an $x$-geodominating set of cardinality $g_x(G)$ is called a $g_x$-set and it is known that it is unique for each vertex $x$. 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