Algebraic fiberings of right-angled Coxeter groups on bipartite cone graphs

We study systems of moves and states in the sense of [JNW21] on certain bipartite graphs, and the associated homomorphisms from the commutator subgroups of right-angled Coxeter groups to $\mathbb Z$. We give examples of right-angled Coxeter groups whose graphs do not satisfy the legal system criterion, but contain finite index right-angled Coxeter subgroups whose graphs admit legal systems. We also provide a family of examples of graphs without legal systems, where no sequence of the considered graph operations yields a graph with a legal system. Moreover, we show that for a bipartite graph and an orbit of non-empty states under the bipartite system of moves, the associated homomorphism from the commutator subgroup of the right-angled Coxeter group to $\mathbb Z$ has finitely generated kernel if and only if the system is legal.