{"paper":{"title":"Parameterized Algorithms on Perfect Graphs for deletion to $(r,\\ell)$-graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Fahad Panolan, Saket Saurabh, Sudeshna Kolay, Venkatesh Raman","submitted_at":"2015-12-14T07:05:04Z","abstract_excerpt":"For fixed integers $r,\\ell \\geq 0$, a graph $G$ is called an {\\em $(r,\\ell)$-graph} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\\ell$ cliques. The class of $(r, \\ell)$ graphs generalizes $r$-colourable graphs (when $\\ell =0)$ and hence not surprisingly, determining whether a given graph is an $(r, \\ell)$-graph is \\NP-hard even when $r \\geq 3$ or $\\ell \\geq 3$ in general graphs.\n  When $r$ and $\\ell$ are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the {\\sc Chromatic Number} problem is solvable in pol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04200","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"}