The labelled reachability problem for undirected graphs where the edges are labelled with elements from a monoid M (more generally, other groupoids or magmas) is known to capture space-bounded complexity classes \(\textsf{L}\) and \({\textsf{NL}}\) . Given a labelled graph G(V, E) (labelled with \(\phi ~ :E \rightarrow M\) ), \(s,t \in V\) and an accepting subset \(F \subseteq M\) , the problem asks to test whether there is a walk P from s to t in G where \(\phi (P) \in F\) . When the accepting element is also part of the input, the problem has been studied by Ramaswamy et al. (2019) [16] for the case when the monoid is aperiodic and the case when the monoid is a group. Motivated by the success in designing space-bounded algorithms for the graph reachability problem in the undirected case, we study the labelled reachability problem when the accepting set is also fixed. This allows us to reveal finer complexity upper and lower bounds, and dichotomies for the problem using the structure of the underlying monoid and the accepting set. In fact, when the monoid M is a group or belongs to the monoid pseudovariety \(\textsf{DA}\) , the previous results imply a deterministic logspace algorithm for the undirected labelled reachability problem over the monoid M for any finite accepting subset as well. We prove the following: To obtain our results, we critically exploit the relationships between Green’s equivalence relations in the union-of-groups monoids and the properties of the product graph (a graph introduced by [16]).