On the structure of solution-sets to regular word equations

Abstract For quadratic word equations, there exists an algorithm based on rewriting rules which generates a directed graph describing all solutions to the equation. regular equations – those for each variable occurs at most once side of equation we investigate properties this graph, such as bounds its diameter, size, and DAG-width, well providing some insights into symmetries in structure. As consequence, obtain combinatorial proof that problem deciding whether has solution is NP.