If a complex is a subcomplex of a diagrammatically reducible 2-complex that has locally indicable fundamental group, then has locally indicable fundamental group.This is a consequence of the Corson-Trace characterization of diagrammatic reducibility.In this paper 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 (LOT) that is diagrammatically reducible of degree 2, and all its quotients are as well, is locally indicable.
Harlander et al. (2026) studied this question.