{"paper":{"title":"Two Results on Outer-String Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Bipartite graphs with a given cyclic order admit a polynomial-time test for constrained outer-string representations based on a single forbidden configuration, while recognizing outer-k-string representations is NP-hard for any fixed k.","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Jan Kratochv\\'il, Peter Stumpf, Todor Anti\\'c, V\\'it Jel\\'inek","submitted_at":"2026-05-12T15:22:54Z","abstract_excerpt":"An \\emph{outer-string representation} of a graph $G$ is an intersection representation of $G$ where vertices are represented by curves (strings) inside the unit disk and each curve has exactly one endpoint on the boundary of the unit disk (the anchor of the curve). Additionally, if each two curves are allowed to cross at most once, we call this an \\emph{outer-$1$-string representation} of $G$. If we impose a cyclic ordering on the vertices of $G$ and require the cyclic order of the anchors to respect this cyclic order, such a representation is called a \\emph{constrained outer-string representa"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For a bipartite graph G (and more generally for any {C3,C5}-free graph G) with a given cyclic order of vertices, we can decide in polynomial time whether G admits a constrained outer-string representation. Our algorithm follows from a characterization by a single forbidden configuration.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The algorithm's correctness rests on the claim that a single forbidden configuration completely characterizes the graphs that admit constrained outer-string representations (abstract, paragraph describing the first result and reference to Biedl et al. [GD 2024]).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Polynomial-time recognition for constrained outer-string representations of {C3,C5}-free graphs with cyclic order, plus NP-hardness for outer-k-string recognition for any fixed k.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Bipartite graphs with a given cyclic order admit a polynomial-time test for constrained outer-string representations based on a single forbidden configuration, while recognizing outer-k-string representations is NP-hard for any fixed k.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d225782dc9def6020a53ec2fe97960a89c66cb672e3956b441976fe52df995ab"},"source":{"id":"2605.12253","kind":"arxiv","version":2},"verdict":{"id":"0fd41dee-d832-4ee6-bdbb-e71bd9fd5e08","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T17:29:51.816601Z","strongest_claim":"For a bipartite graph G (and more generally for any {C3,C5}-free graph G) with a given cyclic order of vertices, we can decide in polynomial time whether G admits a constrained outer-string representation. 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