{"paper":{"title":"Toll number of the strong product of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Polona Repolusk, Tanja Gologranc","submitted_at":"2018-01-23T11:22:39Z","abstract_excerpt":"A tolled walk $T$ between two non-adjacent vertices $u$ and $v$ in a graph $G$ is a walk, in which $u$ is adjacent only to the second vertex of $T$ and $v$ is adjacent only to the second-to-last vertex of $T$. A toll interval between $u,v\\in V(G)$ is a set $T_G(u,v)=\\{x\\in V(G)~|~x \\textrm{ lies on a tolled walk between } u \\textrm{\\, and\\,} v\\}$. A set $S \\subseteq V(G)$ is toll convex, if $T_{G}(u,v)\\subseteq S$ for all $u,v\\in S$. A toll closure of a set $S \\subseteq V(G)$ is the union of toll intervals between all pairs of vertices from $S$. The size of a smallest set $S$ whose toll closur"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.08043","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"}