# Cutting a tree with Subgraph Complementation is hard, except for some small trees

For a property Π, Subgraph Complementation to Π is the problem to find whether there is a subset S of vertices in the input graph G such that modifying G by complementing the subgraph induced by S results in a graph satisfying the property Π. We prove that, the problem of Subgraph Complementation to T-free graphs is NP-Complete, for T being a tree, except for 40 trees of at most 13 vertices (a graph is T-free if it does not contain any induced copies of T). Further, we prove that these hard problems do not admit any subexponential-time algorithms, assuming the Exponential Time Hypothesis. As an additional result, we obtain that Subgraph Complementation to paw-free graphs can be solved in polynomial-time.

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