FPT Algorithms to Compute the Elimination Distance to Bipartite Graphs and More

Abstract

For a hereditary graph class $$\mathcal {H}$$, the $$\mathcal {H}$$-elimination distance of a graph G is the minimum number of rounds needed to reduce G to a member of $$\mathcal {H}$$ by removing one vertex from each connected component in each round. The $$\mathcal {H}$$-treewidth of a graph G is the minimum, taken over all vertex sets X for which each connected component of $$G - X$$ belongs to $$\mathcal {H}$$, of the treewidth of the graph obtained from G by replacing the neighborhood of each component of $$G-X$$ by a clique and then removing $$V(G) \setminus X$$. These parameterizations recently attracted interest because they are simultaneously smaller than the graph-complexity measures treedepth and treewidth, respectively, and the vertex-deletion distance to $$\mathcal {H}$$. For the class $$\mathcal {H}$$ of bipartite graphs, we present non-uniform fixed-parameter tractable algorithms for testing whether the $$\mathcal {H}$$-elimination distance or $$\mathcal {H}$$-treewidth of a graph is at most k. Along the way, we also provide such algorithms for all graph classes $$\mathcal {H}$$ defined by a finite set of forbidden induced subgraphs.