{"paper":{"title":"Algorithmic Complexity of Weakly Semiregular Partitioning and the Representation Number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Ali Dehghan, Arash Ahadi, Mohsen Mollahajiaghaei","submitted_at":"2017-01-20T22:01:22Z","abstract_excerpt":"A graph $G$ is {\\it weakly semiregular} if there are two numbers $a,b$, such that the degree of every vertex is $a$ or $b$. The {\\it weakly semiregular number} of a graph $G$, denoted by $wr(G)$, is the minimum number of subsets into which the edge set of $G$ can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph. We present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two. On the other hand, we show that determining whether $ wr(G) = 2 $ for a given bipartite graph $ G $ with at most three numbers in it"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05934","kind":"arxiv","version":1},"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"}