We introduce the block elimination distance as a measure of how close a graph is to some particular graph class. Formally, given a graph class ${\cal G}$, the class ${\cal B}({\cal G})$ contains all graphs whose blocks belong to ${\cal G}$ and the class ${\cal A}({\cal G})$ contains all graphs where the removal of a vertex creates a graph in ${\cal G}$. Given a hereditary graph class ${\cal G}$, we recursively define ${\cal G}^{(k)}$ so that ${\cal G}^{(0)}={\cal B}({\cal G})$ and, if $k\geq 1$, ${\cal G}^{(k)}={\cal B}({\cal A}({\cal G}^{(k-1)}))$. The block elimination distance of a graph $G$ to a graph class ${\cal G}$ is the minimum $k$ such that $G\in{\cal G}^{(k)}$ and can be seen as an analog of the elimination distance parameter, with the difference that connectivity is now replaced by biconnectivity. We show that, for every non-trivial hereditary class ${\cal G}$, the problem of deciding whether $G\in{\cal G}^{(k)}$ is NP-complete. We focus on the case where ${\cal G}$ is minor-closed and we study the minor obstruction set of ${\cal G}^{(k)}$. We prove that the size of the obstructions of ${\cal G}^{(k)}$ is upper bounded by some explicit function of $k$ and the maximum size of a minor obstruction of ${\cal G}$. This implies that the problem of deciding whether $G\in{\cal G}^{(k)}$ is constructively fixed parameter tractable, when parameterized by $k$. Our results are based on a structural characterization of the obstructions of ${\cal B}({\cal G})$, relatively to the obstructions of ${\cal G}$. We give two graph operations that generate members of ${\cal G}^{(k)}$ from members of ${\cal G}^{(k-1)}$ and we prove that this set of operations is complete for the class ${\cal O}$ of outerplanar graphs. This yields the identification of all members ${\cal O}\cap{\cal G}^{(k)}$, for every $k\in\mathbb{N}$ and every non-trivial minor-closed graph class ${\cal G}$.

中文翻译：

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消除距离

我们介绍了块消除距离，以衡量图形与某些特定图形类别的接近程度。形式上，给定图类$ {\ cal G} $，类$ {\ cal B}（{\ cal G}）$包含其块属于$ {\ cal G} $和类$ {\的所有图。 cal A}（{\ cal G}）$包含所有图形，其中移除顶点会在$ {\ cal G} $中创建图形。给定遗传图类$ {\ cal G} $，我们递归定义$ {\ cal G} ^ {（k）} $，以便$ {\ cal G} ^ {（0）} = {\ cal B}（ {\ cal G}} $，如果$ k \ geq 1 $，则$ {\ cal G} ^ {（k）} = {\ cal B}（{\ cal A}（{\ cal G} ^ {{ k-1）}））$。图$ G $到图类$ {\ cal G} $的块消除距离是最小$ k $，因此$ G \ in {\ cal G} ^ {（k）} $可以看作是消除距离参数的类似物，不同之处在于现在已将连通性替换为双连通性。我们证明 对于每个非平凡的遗传类$ {\ cal G} $，决定$ G \ in {\ cal G} ^ {（k）} $是否是NP完整的问题。我们重点研究$ {\ cal G} $是次要封闭的情况，并研究$ {\ cal G} ^ {（k）} $的次要障碍集。我们证明$ {\ cal G} ^ {（k）} $的障碍物大小由$ k $的某些显式函数和$ {\ cal G} $的较小障碍物的最大大小所限制。这意味着在用$ k $进行参数化时，确定$ G \ in {\ cal G} ^ {（k）} $是否是建设性的固定参数易处理的问题。我们的结果基于相对于$ {\ cal G} $障碍物的$ {\ cal B}（{\ cal G}）$障碍物的结构特征。我们给出了两个图形操作，它们从$ {\ cal G} ^ {（k-1）} $的成员中生成$ {\ cal G} ^ {（k）} $的成员，我们证明这组操作是完整的对于外平面图的类$ {\ cal O} $。这样就可以为每个$ k \ in \ mathbb {N} $和每个非平凡的次要闭合图类标识所有成员$ {\ cal O} \ cap {\ cal G} ^ {（k）} $ $ {\ cal G} $。