Planar Connectivity Models for Cross-Free Societies

Definitions and mathematical statement

Write $(G,\Omega)$ for an object called a society in the supplied record, and let $B=\Omega(G)\subseteq V(G)$ denote its designated boundary vertices, with the order specified by $\Omega$. The record does not define a society or the directed notion of cross-free, and does not specify the permitted class of digraphs. These hypotheses therefore require clarification before a complete mathematical formulation is possible.

The original question asks whether, for every cross-free society $(G,\Omega)$, there exists a planar digraph $H$ such that $B\subseteq V(H)$, with a plane embedding placing the designated boundary vertices on the outer face in the order prescribed by $\Omega$, and preserving exactly the connectivity among those vertices. If $\mathcal C(K,B)$ denotes the intended connectivity data of a digraph $K$ on its designated boundary $B$, the requested preservation condition has the schematic form \[ \mathcal C(H,B)=\mathcal C(G,B). \] Here $\mathcal C$ is an unresolved placeholder: the supplied record does not say whether connectivity means directed reachability, simultaneous disjoint directed paths, or another relation. No particular interpretation is imposed in this draft.

The original statement refers to the outer-face order of $\Omega(H)$, although only $\Omega(G)$ and $\Omega$ have been introduced. Interpreting this as the same designated boundary $B$, carrying its prescribed order, is a plausible intended reading but requires confirmation. The cleaned statement asks more generally for a characterization of the cross-free societies admitting such a planar connectivity model. Whether every cross-free society admits one is the explicit universal existence question in the original excerpt; the supplied record does not establish its answer.

Short English statement

Determine whether every cross-free society has a planar directed graph preserving exactly its boundary connectivity and prescribed boundary order; more generally, characterize those that do. The directed cross-free condition, the meaning of connectivity, and the boundary notation require clarification before this becomes a fully specified mathematical problem.

Sources

• Local UnsolvedMath registry: permanent ID $30004984$, source problem code catalogue entry, title Planar Connectivity Models for Cross-Free Societies, in the supplied Oberwolfach Reports Open Problems collection. No proposer is identified for this question; the proposed year is recorded as 2022.
• Supplied source citation: Graph Theory (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/1, pp. 5--78. Supplied source URL and literature reference, labeled Oberwolfach Report 1/2022: https://doi.org/10.4171/owr/2022/1. The citation's 2023 label and its 2022 report designation are retained without resolving the discrepancy.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the question open, describes it as asking for an exact planar connectivity-preserving replacement, and reports that no authoritative resolution was found. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports comparison with the official report. These are supplied source-verification claims; this draft does not independently verify them or the external source.