arXiv2026
The satisfiability problem for word equations asks whether variables can be replaced by words so that the two sides become equal. For regular word equations, in which each variable occurs at most once on each side, satisfiability is NP-complete. For general quadratic word equations, in which each variable occurs at most twice in total, satisfiability is NP-hard, but its membership in NP remains open. We consider an intermediate class: quadratic word equations with a linear side, where each variable occurs at most once on one designated side. We show that the Nielsen graph of an equation $U=V$ in this class, with total length $N=|U|+|V|$, has diameter $O(N^{12})$, measured over reachable pairs of vertices. Together with the known NP-hardness for regular word equations, this result establishes NP-completeness of satisfiability for this class. We also show that each strongly connected component is isomorphic to the length-preserving reachability graph of a regular equation, and that the condensation graph has depth at most $|U|+2|V|$.For arbitrary quadratic word equations, we further show that Nielsen paths can be shortened to a length polynomial in the equation size and the number of prefix operations between variables repeated on opposite sides. This yields NP membership for subclasses defined by variable occurrence arrangements.