A well-established research line in structural and algorithmic graph theory is characterizing graph classes by listing their minimal obstructions. When this list is finite for some class $\mathcal C$ we obtain a polynomial-time algorithm for recognizing graphs in $\mathcal C$, and from a logic point of view, having finitely many obstructions corresponds to being definable by a universal sentence. However, in many cases we study classes with infinite sets of minimal obstructions, and this might have neither algorithmic nor logic implications for such a class. Some decades ago, Skrien (1982) and Damaschke (1990) introduced finite expressions of graph classes by means of forbidden orientations and forbidden linear orderings, and recently, similar research lines appeared in the literature, such as expressions by forbidden circular orders, by forbidden tree-layouts, and by forbidden edge-coloured graphs. In this paper, we introduce local expressions of graph classes; a general framework for characterizing graph classes by forbidden equipped graphs. In particular, it encompasses all research lines mentioned above, and we provide some new examples of such characterizations. Moreover, we see that every local expression of a class $\mathcal C$ yields a polynomial-time certification algorithm for graphs in $\mathcal C$. Finally, from a logic point of view, we show that being locally expressible corresponds to being definable in the logic SNP introduced by Feder and Vardi (1999).

翻译：结构图论与算法图论中一个成熟的研究方向是通过列出图类的最小障碍集来刻画图类。当某图类$\mathcal C$的障碍集有限时，我们可获得识别该图类中图的多项式时间算法；从逻辑角度看，拥有有限障碍集对应于可被全称语句定义。然而，许多情况下我们研究的图类具有无限的最小障碍集，这对该图类既无算法意义也无逻辑意义。数十年前，Skrien（1982年）与Damaschke（1990年）通过禁止定向与禁止线性序引入了图类的有限表达，近期文献中出现了类似研究方向，例如通过禁止循环序、禁止树布局和禁止边着色图进行表达。本文引入图类的局部表达——一种通过禁止配备图来刻画图类的通用框架。该框架特别涵盖了上述所有研究方向，并提供了此类刻画的新实例。此外，我们发现每个图类$\mathcal C$的局部表达均可为$\mathcal C$中的图生成多项式时间认证算法。最后，从逻辑角度，我们证明局部可表达性对应于Feder与Vardi（1999年）引入的SNP逻辑中的可定义性。