{"paper":{"title":"Counting $4\\times 4$ Matrix Partitions of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"David Richerby, Leslie Ann Goldberg, Martin Dyer","submitted_at":"2014-07-29T17:54:05Z","abstract_excerpt":"Given a symmetric matrix $M\\in \\{0,1,*\\}^{D\\times D}$, an $M$-partition of a graph $G$ is a function from $V(G)$ to $D$ such that no edge of $G$ is mapped to a $0$ of $M$ and no non-edge to a $1$. We give a computer-assisted proof that, when $|D|=4$, the problem of counting the $M$-partitions of an input graph is either in FP or is #P-complete. Tractability is proved by reduction to the related problem of counting list $M$-partitions; intractability is shown using a gadget construction and interpolation. We use a computer program to determine which of the two cases holds for all but a small nu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.7799","kind":"arxiv","version":4},"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"}