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 $\mathscr{G} = \langle \mathscr{G}_{t} \rangle_{t \in \mathbb{N}}$ 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 $\mathsf{p}$ if there exists a function $f \colon \mathbb{N} \to \mathbb{N}$ such that, for every $k \in \mathbb{N}$ and every graph $G$, 1) if $\mathsf{p}(G) \leq k$, then for every $\mathscr{G} \in \mathfrak{G},$ $\mathscr{G}_{f(k)} \not\leqslant G$, and 2) if for every $\mathscr{G} \in \mathfrak{G},$ $\mathscr{G}_{k} \not\leqslant G$, then $\mathsf{p}(G) \leq f(k).$ 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.

翻译：我们建立了一个参数化框架，用于获得图参数关于图类上拟序$\leqslant$的障碍刻画。该框架的核心概念是$\leqslant$-参数图：一个由非负整数索引的非$\leqslant$递减图序列$\mathscr{G} = \langle \mathscr{G}_{t} \rangle_{t \in \mathbb{N}}$。参数图使我们能够定义捕获图参数近似行为的组合对象。若存在函数$f \colon \mathbb{N} \to \mathbb{N}$使得对任意$k \in \mathbb{N}$和任意图$G$满足：1) 当$\mathsf{p}(G) \leq k$时，对每个$\mathscr{G} \in \mathfrak{G}$都有$\mathscr{G}_{f(k)} \not\leqslant G$；2) 当对每个$\mathscr{G} \in \mathfrak{G}$都有$\mathscr{G}_{k} \not\leqslant G$时，则$\mathsf{p}(G) \leq f(k)$，则称有限$\leqslant$-参数图集$\mathfrak{G}$为参数$\mathsf{p}$的$\leqslant$-通用障碍。为巩固我们的观点，我们确定了保证通用障碍存在的充分序论条件，并在此情况下研究了其对固定参数可解算法存在的算法影响。我们的参数化框架对图类性质的有限障碍刻画具有进一步意义。$\leqslant$-类性质定义为在集合包含下封闭的任意$\leqslant$-封闭图类集合。通过将我们的参数化框架与序论中的既定结果相结合，我们推导出精确的序论刻画，确保$\leqslant$-类性质可以用排除有限个$\leqslant$-参数图来描述。