Block Elimination Distance

We introduce the parameter of block elimination distance as a measure of how close a graph is to some particular graph class. Formally, given a graph class G, the class B(G) contains all graphs whose blocks belong to G and the class A(G) contains all graphs where the removal of a vertex creates a graph in G. Given a hereditary graph class G, we recursively define G^{( k )} so that G^{(0 )}= B(G) and, if k≥ 1, G^{( k )}= B(A(G^{( k - 1 )}) ). The block elimination distance of a graph G to a graph class G is the minimum k such that G∈ G^{( k )} and can be seen as an analog of the elimination distance parameter, defined in [J. Bulian & A. Dawar. Algorithmica, 75(2):363–382, 2016], with the difference that connectivity is now replaced by biconnectivity. We show that, for every non-trivial hereditary class G, the problem of deciding whether G∈ G^{( k )} is NP-complete. We focus on the case where G is minor-closed and we study the minor obstruction set of G^{( k )} i.e., the minor-minimal graphs not in G^{( k )}. We prove that the size of the obstructions of G^{( k )} is upper bounded by some explicit function of k and the maximum size of a minor obstruction of G. This implies that the problem of deciding whether G∈ G^{( k )} is constructively fixed parameter tractable, when parameterized by k. Our results are based on a structural characterization of the obstructions of B(G), relatively to the obstructions of G. Finally, we give two graph operations that generate members of G^{( k )} from members of G^{( k - 1 )} and we prove that this set of operations is complete for the class O of outerplanar graphs. This yields the identification of all members O∩ G^{( k )}, for every k∈ N and every non-trivial minor-closed graph class G.