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The $n$-barbell graph $Bar_n$ with $2n$ vertices, is formed by joining two copies of a complete graph $K_n$ by a single edge. We prove that for every $n\\geq 2$, $Bar_n$ is not $\\mathcal{D}$-unique, that is, there is another non-isomorphic graph with the same domination polynomial. More precisely, we show that for every $n$, the $\\mathcal{D}$-equivalence class of barbell graph, $[Bar_n]$, contains many g"},"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":"1511.00159","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-31T18:04:43Z","cross_cats_sorted":[],"title_canon_sha256":"3ab80d201fcaab52f93bd37e27e92804625b6f176d671ab6cc50ca3c05f665a3","abstract_canon_sha256":"10c3d70d9a19c46ea2e4148ba72d55f00794b87e17bb0b963ac6ceef407931e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:33.526416Z","signature_b64":"00bIHnaENhGQ9wBiUec7ep4Fayh74/I5elWXoiHHX+W6arp8f5e68KLVeKXhuq8PCwRUrDNiqXUkxofZ9bWrBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e79524193252af6a4ac4e53b39860b5d7e4513696bfe61b16e626bc68013255","last_reissued_at":"2026-05-18T01:26:33.525667Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:33.525667Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On $\\mathcal{D}$-equivalence classes of some graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Saeid Alikhani, Somayeh Jahari","submitted_at":"2015-10-31T18:04:43Z","abstract_excerpt":"Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. The $n$-barbell graph $Bar_n$ with $2n$ vertices, is formed by joining two copies of a complete graph $K_n$ by a single edge. We prove that for every $n\\geq 2$, $Bar_n$ is not $\\mathcal{D}$-unique, that is, there is another non-isomorphic graph with the same domination polynomial. More precisely, we show that for every $n$, the $\\mathcal{D}$-equivalence class of barbell graph, $[Bar_n]$, contains many g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.00159","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1511.00159","created_at":"2026-05-18T01:26:33.525808+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.00159v2","created_at":"2026-05-18T01:26:33.525808+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.00159","created_at":"2026-05-18T01:26:33.525808+00:00"},{"alias_kind":"pith_short_12","alias_value":"PZ4VEQMTEUVP","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_16","alias_value":"PZ4VEQMTEUVPNJFM","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_8","alias_value":"PZ4VEQMT","created_at":"2026-05-18T12:29:37.295048+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PZ4VEQMTEUVPNJFMJZJ3HGDAWX","json":"https://pith.science/pith/PZ4VEQMTEUVPNJFMJZJ3HGDAWX.json","graph_json":"https://pith.science/api/pith-number/PZ4VEQMTEUVPNJFMJZJ3HGDAWX/graph.json","events_json":"https://pith.science/api/pith-number/PZ4VEQMTEUVPNJFMJZJ3HGDAWX/events.json","paper":"https://pith.science/paper/PZ4VEQMT"},"agent_actions":{"view_html":"https://pith.science/pith/PZ4VEQMTEUVPNJFMJZJ3HGDAWX","download_json":"https://pith.science/pith/PZ4VEQMTEUVPNJFMJZJ3HGDAWX.json","view_paper":"https://pith.science/paper/PZ4VEQMT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.00159&json=true","fetch_graph":"https://pith.science/api/pith-number/PZ4VEQMTEUVPNJFMJZJ3HGDAWX/graph.json","fetch_events":"https://pith.science/api/pith-number/PZ4VEQMTEUVPNJFMJZJ3HGDAWX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PZ4VEQMTEUVPNJFMJZJ3HGDAWX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PZ4VEQMTEUVPNJFMJZJ3HGDAWX/action/storage_attestation","attest_author":"https://pith.science/pith/PZ4VEQMTEUVPNJFMJZJ3HGDAWX/action/author_attestation","sign_citation":"https://pith.science/pith/PZ4VEQMTEUVPNJFMJZJ3HGDAWX/action/citation_signature","submit_replication":"https://pith.science/pith/PZ4VEQMTEUVPNJFMJZJ3HGDAWX/action/replication_record"}},"created_at":"2026-05-18T01:26:33.525808+00:00","updated_at":"2026-05-18T01:26:33.525808+00:00"}