{"paper":{"title":"On the existence of $3$- and $4$-kernels in digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"C\\'esar Hern\\'andez-Cruz, Sebasti\\'an Gonz\\'alez Hermosillo de la Maza","submitted_at":"2016-10-18T18:27:08Z","abstract_excerpt":"Let $D = (V(D), A(D))$ be a digraph. A subset $S \\subseteq V(D)$ is $k$-independent if the distance between every pair of vertices of $S$ is at least $k$, and it is $\\ell$-absorbent if for every vertex $u$ in $V(D) \\setminus S$ there exists $v \\in S$ such that the distance from $u$ to $v$ is less than or equal to $\\ell$. A $k$-kernel is a $k$-independent and $(k-1)$-absorbent set. A kernel is simply a $2$-kernel.\n  A classical result due to Duchet states that if every directed cycle in a digraph $D$ has at least one symmetric arc, then $D$ has a kernel. We propose a conjecture generalizing thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.05730","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"}