Block Elimination Distance

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 ${\cal G}$.