For $N>1$, we constructed a canonical connected fundamental domain for $\Gamma _0(N)$ in Nie and Parent (2024, Connected fundamental domains for congruence subgroups), utilizing an interesting function $W: {\mathbb Z}/N\to {{\mathbb N}}$. In this article, we further study the function W, prove some identities, and use it to match the cusps, with widths, produced by our connected fundamental domain with the known cusp classes of $\Gamma _0(N)$. Furthermore, we list the boundary arcs and the gluing patterns of our connected fundamental domain, a key step in understanding the modular curve $X_0(N)$ by this approach.

If a complex X is a subcomplex of a diagrammatically reducible (DR) 2-complex Y that has locally indicable fundamental group, then X has locally indicable fundamental group. This is a consequence of the Corson–Trace characterization of diagrammatic reducibility. In this article, we use a Corson–Trace-like characterization of diagrammatic reducibility away from a subcomplex to obtain a considerable stronger result. We apply this to the question of local indicability in the context of Whitehead’s asphericity conjecture. We show that an injective labeled oriented tree that is DR of degree 2, and all its quotients are as well, is locally indicable.

Diagrammatic reducibility DR and its generalization, vertex asphericity VA, are combinatorial tools developed for detecting asphericity of a 2-complex. Here we present tests for a relative version of VA that apply to pairs of 2-complexes $(L,K)$, where K is a subcomplex of L. We show that a relative weight test holds for injective labeled oriented trees, implying that they are VA and hence aspherical. This strengthens a result obtained by the authors in 2017 and simplifies the original proof.