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A subset of vertices $D$ is said to be a $\\tau$-dynamic monopoly, if $V(G)$ can be partitioned into subsets $D_0, D_1, \\ldots, D_k$ such that $D_0=D$ and for any $i\\in \\{0, \\ldots, k-1\\}$, each vertex $v$ in $D_{i+1}$ has at least $\\tau(v)$ neighbors in $D_0\\cup \\ldots \\cup D_i$. Denote the size of smallest $\\tau$-dynamic monopoly by $dyn_{\\tau}(G)$ and the average of thresholds in $\\tau$ by $\\overline{\\tau}$. We show that the values of $dyn_{\\tau}(G)$ over all assignments $\\tau$ with the s"},"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":"1405.6138","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-23T17:38:31Z","cross_cats_sorted":[],"title_canon_sha256":"a716f7f6190582446886d0ab1c07de21c4a13795b4e30feaee7d011fc64dedd3","abstract_canon_sha256":"e751e23aad8813b53e87cb4361e701e1f97941718c99835925bf3dad50c3a4d1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:51:14.012105Z","signature_b64":"kZSFlcpqSGqjc7cc6wjs2uI1KQJwe7pRNtvtVL2FZYthAauyAnfJGxBvYCrln6FH21v5ALkNCgpEyV3DGyJ2Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"16eaae2cb8371ec1b31ce2d9bab4b9b5e62f8a06e292ab5b06b5f0d0bae109ff","last_reissued_at":"2026-05-18T02:51:14.011388Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:51:14.011388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the largest dynamic monopolies of graphs with a given average threshold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kaveh Khoshkhah, Manouchehr Zaker","submitted_at":"2014-05-23T17:38:31Z","abstract_excerpt":"Let $G$ be a graph and $\\tau$ be an assignment of nonnegative integer thresholds to the vertices of $G$. A subset of vertices $D$ is said to be a $\\tau$-dynamic monopoly, if $V(G)$ can be partitioned into subsets $D_0, D_1, \\ldots, D_k$ such that $D_0=D$ and for any $i\\in \\{0, \\ldots, k-1\\}$, each vertex $v$ in $D_{i+1}$ has at least $\\tau(v)$ neighbors in $D_0\\cup \\ldots \\cup D_i$. Denote the size of smallest $\\tau$-dynamic monopoly by $dyn_{\\tau}(G)$ and the average of thresholds in $\\tau$ by $\\overline{\\tau}$. 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