Switching Classes: Characterization and Computation

TL;DR

It is shown that the recognition of the superclass is NP-complete for $H$-free graphs when $H$ is a sufficiently long path or cycle, and it cannot be solved in subexponential time assuming the Exponential Time Hypothesis.

Abstract

In a graph, the switching operation reverses adjacencies between a subset of vertices and the others. For a hereditary graph class $\mathcal{G}$, we are concerned with the maximum subclass and the minimum superclass of $\mathcal{G}$ that are closed under switching. We characterize the maximum subclass for many important classes $\mathcal{G}$, and prove that it is finite when $\mathcal{G}$ is minor-closed and omits at least one graph. For several graph classes, we develop polynomial-time algorithms to recognize the minimum superclass. We also show that the recognition of the superclass is NP-complete for $H$-free graphs when $H$ is a sufficiently long path or cycle, and it cannot be solved in subexponential time assuming the Exponential Time Hypothesis.

Figures (14)

• Figure 1: small graphs.
• Figure 2: Switching equivalent graphs of $C_{4}$ and $C_{5}$. Switching the solid nodes (or the rest) results in the first graph in the list.
• Figure 3: The Hasse diagram of graph classes studied in Section \ref{['sec:lower']}.
• Figure 4: Switching equivalent graphs of $C_{6}$. The set $A$ consists of all the empty nodes or all the solid nodes.
• Figure 5: All five-vertex graph containing a $C_{4}$ form three groups. The set $A$ consists of the solid nodes.

Theorems & Definitions (84)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1: folklore
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Lemma 3.1
• Lemma 3.2