We establish a parametric framework for obtaining obstruction characterizations of graph parameters with respect to a quasi-ordering \(\leqslant \) on graphs. At the center of this framework lies the concept of a \(\leqslant \) -parametric graph: a non \(\leqslant \) -decreasing sequence \(\mathcal {G} = \langle \mathcal {G}_{t} \rangle _{t \in \mathbb {N}}\) of graphs indexed by non-negative integers. Parametric graphs allow us to define combinatorial objects that capture the approximate behaviour of graph parameters. A finite set \(\mathfrak {G}\) of \(\leqslant \) -parametric graphs is a \(\leqslant \) -universal obstruction for a parameter \(\textsf{p}\) if there exists a function \(f :\mathbb {N}\rightarrow \mathbb {N}\) such that, for every \(k \in \mathbb {N}\) and every graph G, 1) if \(\textsf{p}(G) \le k\) , then for every \(\mathcal {G} \in \mathfrak {G},\) \(\mathcal {G}_{f(k)} \not \leqslant G\) , and 2) if for every \(\mathcal {G} \in \mathfrak {G},\) \(\mathcal {G}_{k} \not \leqslant G\) , then \(\textsf{p}(G) \le f(k).\) To solidify our point of view, we identify sufficient order-theoretic conditions that guarantee the existence of universal obstructions and in this case we examine algorithmic implications on the existence of fixed-parameter tractable algorithms. Our parametric framework has further implications related to finite obstruction characterizations of properties of graph classes. A \(\leqslant \) -class property is defined as any set of \(\leqslant \) -closed graph classes that is closed under set inclusion. By combining our parametric framework with established results from order theory, we derive a precise order-theoretic characterization that ensures \(\leqslant \) -class properties can be described in terms of the exclusion of a finite set of \(\leqslant \) -parametric graphs.
