We investigate a structural generalisation of treewidth we call $\mathcal{A}$-blind-treewidth where $\mathcal{A}$ denotes an annotated graph class. This width parameter is defined by evaluating only the size of those bags $B$ of tree-decompositions for a graph $G$ where ${(G,B) \notin \mathcal{A}}$. For the two cases where $\mathcal{A}$ is (i) the class $\mathcal{B}$ of all pairs ${(G,X)}$ such that no odd cycle in $G$ contains more than one vertex of ${X \subseteq V(G)}$ and (ii) the class $\mathcal{B}$ together with the class $\mathcal{P}$ of all pairs ${(G,X)}$ such that the "torso" of $X$ in $G$ is planar. For both classes, $\mathcal{B}$ and ${\mathcal{B} \cup \mathcal{P}}$, we obtain analogues of the Grid Theorem by Robertson and Seymour and FPT-algorithms that either compute decompositions of small width or correctly determine that the width of a given graph is large. Moreover, we present FPT-algorithms for Maximum Independent Set on graphs of bounded $\mathcal{B}$-blind-treewidth and Maximum Cut on graphs of bounded ${(\mathcal{B}\cup\mathcal{P})}$-blind-treewidth.

翻译：我们研究一种称为$\mathcal{A}$-盲树宽度的树宽度的结构推广，其中$\mathcal{A}$表示一个带标注的图类。该宽度参数仅通过评估图$G$的树分解中那些满足${(G,B) \notin \mathcal{A}}$的袋$B$的大小来定义。针对以下两种情况：(i) $\mathcal{A}$为类$\mathcal{B}$，包含所有满足$G$中不存在奇环包含${X \subseteq V(G)}$中多于一个顶点的对${(G,X)}$；(ii) $\mathcal{A}$为类$\mathcal{B}$与类$\mathcal{P}$的并集，其中$\mathcal{P}$包含所有满足$X$在$G$中的"躯干"是平面的对${(G,X)}$。对于这两个类$\mathcal{B}$和${\mathcal{B} \cup \mathcal{P}}$，我们获得了Robertson与Seymour的网格定理的类比，以及固定参数可处理（FPT）算法，这些算法要么计算小宽度的分解，要么正确判定给定图的宽度较大。此外，我们针对有界$\mathcal{B}$-盲树宽度的图提出了最大独立集问题的FPT算法，以及针对有界${(\mathcal{B}\cup\mathcal{P})}$-盲树宽度的图提出了最大割问题的FPT算法。