Move Classification of Countable Vertex-Finite Graphs

Definitions and mathematical statement

Let $\mathcal G$ denote the class described in the supplied record as countable vertex-finite graphs. The record does not specify the graph conventions or define vertex-finiteness, so these conventions remain unresolved.

The supplied passage asserts the following eight conjectural equalities for $\mathcal G$. The three-digit labels and the right-hand strings below are reproduced literally; in particular, the letters $h$ and $i$ have not been silently replaced by mathematical delimiters. \[ \begin{aligned} \text{(0)}\quad \text{000} &= \text{h(O), (I-), (R+), (S), (C+), (P+), (K+)i},\\ \text{(1)}\quad \text{001} &= \text{h(O), (I-), (R+), (S)i},\\ \text{(2)}\quad \text{010} &= \text{h(O), (I-), (K+)i},\\ \text{(3)}\quad \text{011} &= \text{h(O), (I-)i},\\ \text{(4)}\quad \text{100} &= \text{h(O), (I+), (R+), (C+), (P+), (K+)i},\\ \text{(5)}\quad \text{101} &= \text{h(O), (I+), (R+)i},\\ \text{(6)}\quad \text{110} &= \text{h(O), (I+), (K+)i},\\ \text{(7)}\quad \text{111} &= \text{h(O), (I+)i}. \end{aligned} \] The intended problem is to establish the eight move-classification assertions represented by this passage. However, the supplied record does not define the eight equivalence relations represented by the three-digit labels, the moves $(O),(I-),(I+),(R+),(S),(C+),(P+),(K+)$, their applicability conditions, or the operation represented by the right-hand notation. Thus the display preserves the asserted classification data but does not constitute a self-contained mathematical conjecture. In particular, its precise quantifiers, graph hypotheses, and interpretation as equalities of relations cannot be certified from the supplied evidence.

Short English statement

The supplied conjecture proposes eight classifications of countable vertex-finite graphs using specified lists of moves. The extracted record omits the definitions needed to interpret these classifications, so a precise self-contained statement remains unrecoverable. Its supplied open status requires independent verification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005845$, title: Move Classification of Countable Vertex-Finite Graphs; exact source problem code: catalogue entry. The record attributes the problem to the Oberwolfach Reports collection, gives proposed year 2024, and supplies no individual proposer.
• Combinatorial $*$-algebras (2024), Oberwolfach Reports, DOI 10.4171/owr/2024/12, pp. 731--788. Supplied report reference: https://doi.org/10.4171/owr/2024/12. The collection describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Oberwolfach Report 12/2024, https://oa.tib.eu/renate/server/api/core/bitstreams/24eefa01-308b-4393-94b5-1598eccca3dd/content.
• Supplied literature source: Graded K-theory and Leavitt path algebras, https://doi.org/10.1007/s10801-022-01184-5.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, describes all eight classifications as conjectures, and reports substantial special-case classification results but no proof of the full countable vertex-finite graph list in the literature checked. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The supplied statement-verification note says that the official report was checked and confirms a genuine eight-part move-classification conjecture, but that the extraction omits the definitions of the eight equivalence relations. It marks the statement unrecoverable, promotes no separate clean statement, and retains the original public record for provenance. These are supplied verification claims, not source visits performed for this draft. The registry record was updated on 2026-09-16T12:06:53Z.