Nuclear dimension for graph C*-algebras with E1 having a cycle with exit and E0 having a cycle without exit

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 the subgraph E1 contains a cycle with an exit and the subgraph E0 contains a cycle without an exit.

Background

The authors analyze finite graphs E with exactly one nontrivial gauge-invariant ideal, arising from a single saturated hereditary subgraph E1 and complementary subgraph E0. They compile known values of the nuclear dimension of C*(E) depending on whether E1 and E0 each contain (i) a cycle with an exit, (ii) a cycle without an exit, or (iii) no cycles.

In this classification, several combinations are resolved with nuclear dimension 0 or 1 via existing results, but the case where the ideal component E1 has a cycle with an exit while the quotient component E0 has a cycle without an exit remains unresolved, as indicated in their summary table.