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For $X \\subset V(G)$, the difference of $X$, $d(X) := |X| - |N (X)|$ where $N(X)$ is the neighborhood of $X$ and $\\max \\, \\{d(X):X\\subset V(G)\\}$ is called the critical difference of $G$. $X$ is called a critical set if $d(X)$ equals the critical difference and ker$(G)$ is the intersection of all critical sets. It is known that ker$(G)$ is an independent (vertex) set of $G$. diadem$(G)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with $d(S) > 0$ if no proper subset of $S$ has positive difference.\n  A graph "},"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":"1701.03040","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-11T16:17:37Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"779aeaa0ae106942547bad5a5b1e52663b650fd1064b3dc3fb03462f52fa9a5a","abstract_canon_sha256":"4f8bf9aaba17d0a95c27d530546ee274ea56795640621427eef459c95356a55c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:40.031697Z","signature_b64":"xCOLAS38vRpwRaP2OdR8Y1oIjy9S3nMKQVQNXgzSiyz2gt9dBnru3HLEC3fYiYyg4Sc4xsMNzY6O+Av4h9x9Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66d6f865703a6f703569acfd2bea66ac6ee50c99d4704fc3bc4ce5e0cbaffd6e","last_reissued_at":"2026-05-18T00:20:40.030950Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:40.030950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Problems on Matchings and Independent Sets of a Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Amitava Bhattacharya, Anupam Mondal, T. Srinivasa Murthy","submitted_at":"2017-01-11T16:17:37Z","abstract_excerpt":"Let $G$ be a finite simple graph. For $X \\subset V(G)$, the difference of $X$, $d(X) := |X| - |N (X)|$ where $N(X)$ is the neighborhood of $X$ and $\\max \\, \\{d(X):X\\subset V(G)\\}$ is called the critical difference of $G$. $X$ is called a critical set if $d(X)$ equals the critical difference and ker$(G)$ is the intersection of all critical sets. It is known that ker$(G)$ is an independent (vertex) set of $G$. diadem$(G)$ is the union of all critical independent sets. 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