A Note on The Complexity of Graph Parameters and The Uniqueness of their Realizations

Let \( \nu \) be some graph parameter and let \( \mathcal{G} \) be a class of graphs for which \( \nu \) can be computed in polynomial time. In this situation, it is often possible to devise a strategy to decide in polynomial time whether \( \nu \) has a unique realization for some graph in \( \mathcal{G} \). We first give an informal description of the conditions that allow one to devise such a strategy, and then we demonstrate our approach for three well-known graph parameters: the domination number, the independence number, and the chromatic number.