Block Elimination Distance
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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 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, 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). 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. 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.
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