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Nuclear dimension for graph C*-algebras with both E1 and E0 having cycles without exits

Determine the nuclear dimension of the graph C*-algebra C*(E) when E is a finite directed graph with a single saturated hereditary subgraph E1 and complementary subgraph E0 (so C*(E) has a single nontrivial gauge-invariant ideal), under the configuration where both E1 and E0 contain cycles without exits.

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Background

Within the same classification framework for finite graphs with a single gauge-invariant ideal, the authors present a table summarizing known nuclear dimension values for C*(E) across all combinations of cycle types in E1 and E0. Many cases are settled by prior work or by results in this paper (yielding nuclear dimension 0 or 1).

However, when both the ideal component E1 and the quotient component E0 each contain cycles without exits, the nuclear dimension remains undetermined, as highlighted by the authors.

References

So we obtain the following table of values for the nuclear dimension of C*(E) (a question mark indicates that the value is unknown), depending on whether each of the two components contains a cycle with an exit, contains a cycle without an exit, or contains no cycles. Minimal examples of the two unknown cases are illustrated below.

— Nuclear dimension of extensions of commutative C*-algebras by Kirchberg algebras (2409.12872 - Evington et al., 19 Sep 2024) in Remark 5.3, Section 5