一句话：给出构造性元定理，可在参数化时间内判定图是否存在满足 CMSO 性质且有界树宽的母图，并用于图补全问题。
本文研究图补全问题：给定 CMSO 句子 φ、整数 t 以及最大度数至多 Δ 的连通图 G，判定是否存在包含 G 且树宽至多 t 的母图 G′ 使 G′ ⊨ φ。作者通过构造性证明给出算法 A，其运行时间为 f(|φ|,t)·2^{O(Δ·t)}·|G|^{O(t)}，其中 f 为可计算函数。作为应用，该元定理可判定 G 是否存在直径至多 d 的平面母图，并证明固定 k 时 k-外平面母图直径判定关于 d 强一致固定参数可处理。结果还可推广：直径参数可替换为任意收缩封闭且有效 CMSO 可定义的参数（如顶点覆盖数、支配数等），平面性可放宽到有界亏格与有界局部树宽。
Let CMSO denote the counting monadic second order logic of graphs. We give a constructive proof that for some computable function $f$, there is an algorithm $\mathfrak{A}$ that takes as input a CMSO sentence $\varphi$, a positive integer $t$, and a connected graph $G$ of maximum degree at most $\Delta$, and determines, in time $f(|\varphi|,t)\cdot 2^{O(\Delta \cdot t)}\cdot |G|^{O(t)}$, whether $G$ has a supergraph $G'$ of treewidth at most $t$ such that $G'\models \varphi$. The algorithmic metatheorem described above sheds new light on certain unresolved questions within the framework of graph completion algorithms. In particular, using this metatheorem, we provide an explicit algorithm that determines, in time $f(d)\cdot 2^{O(\Delta \cdot d)}\cdot |G|^{O(d)}$, whether a connected graph of maximum degree $\Delta$ has a planar supergraph of diameter at most $d$. Additionally, we show that for each fixed $k$, the problem of determining whether $G$ has an $k$-outerplanar supergraph of diameter at most $d$ is strongly uniformly fixed parameter tractable with respect to the parameter $d$. This result can be generalized in two directions. First, the diameter parameter can be replaced by any contraction-closed effectively CMSO-definable parameter $\mathbf{p}$. Examples of such parameters are vertex-cover number, dominating number, and many other contraction-bidimensional parameters. In the second direction, the planarity requirement can be relaxed to bounded genus, and more generally, to bounded local treewidth.