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

TL;DR

This paper investigates the computational complexity of Subgraph Complementation problems related to tree-free graphs, proving NP-Completeness for most cases, identifying some polynomial cases, and establishing hardness results under ETH.

Contribution

It classifies the complexity of Subgraph Complementation to T-free graphs for all trees T, showing NP-Completeness except for 41 small trees, and introduces polynomial-time solutions for paw-free graphs.

Findings

NP-Completeness for most T-free graphs when T is a tree

Polynomial-time solvability for paw-free graphs

No subexponential algorithms under ETH for these problems

Abstract

For a graph property $\Pi$, Subgraph Complementation to $\Pi$ is the problem to find whether there is a subset $S$ of vertices of the input graph $G$ such that modifying $G$ by complementing the subgraph induced by $S$ results in a graph satisfying the property $\Pi$. We prove that the problem of Subgraph Complementation to $T$-free graphs is NP-Complete, for $T$ being a tree, except for 41 trees of at most 13 vertices (a graph is $T$-free if it does not contain any induced copies of $T$). This result, along with the 4 known polynomial-time solvable cases (when $T$ is a path on at most 4 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.