Cutting a tree with subgraph complementation is hard, except for some small trees

Abstract

AbstractFor a graph property , Subgraph Complementation to is the problem to find whether there is a subset of vertices of the input graph such that modifying by complementing the subgraph induced by results in a graph satisfying the property . We prove that the problem of Subgraph Complementation to ‐free graphs is NP‐Complete, for being a tree, except for 41 trees of at most 13 vertices (a graph is ‐free if it does not contain any induced copies of ). This result, along with the four known polynomial‐time solvable cases (when is a path on at most four vertices), leaves behind 37 open cases. 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.