{"paper":{"title":"A dichotomy for the kernel by $H$-walks problem 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, Hortensia Galeana-S\\'anchez","submitted_at":"2016-05-31T12:02:36Z","abstract_excerpt":"Let $H = (V_H, A_H)$ be a digraph which may contain loops, and let $D = (V_D, A_D)$ be a loopless digraph with a coloring of its arcs $c: A_D \\to V_H$. An $H$-walk of $D$ is a walk $(v_0, \\dots, v_n)$ of $D$ such that $(c(v_{i-1}, v_i), c(v_i, v_{i+1}))$ is an arc of $H$, for every $1 \\le i \\le n-1$. For $u, v \\in V_D$, we say that $u$ reaches $v$ by $H$-walks if there exists an $H$-walk from $u$ to $v$ in $D$. A subset $S \\subseteq V_D$ is a kernel by $H$-walks of $D$ if every vertex in $V_D \\setminus S$ reaches by $H$-walks some vertex in $S$, and no vertex in $S$ can reach another vertex in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.09589","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"}