Dichotomy results for classes of countable graphs
Vittorio Cipriani, Ekaterina Fokina, Matthew Harrison-Trainor, Liling Ko, Dino Rossegger
TL;DR
The paper analyzes classes of countable graphs formed by forbidding a single finite induced subgraph, revealing a sharp dichotomy tied to the four-vertex path. If the forbidden graph is not contained in the infinite P4-free family, the resulting class ison top for effective bi-interpretability, capturing the full spectrum of computable structural behaviors; if the forbidden graph is a subgraph of P4, the class is structurally simple with computable embeddability and Sigma-smallness. The authors connect infinite-case phenomena to the classical finite dichotomy around P4, and develop a descriptive-set-theoretic framework (Borel reducibility, BBF-equivalence) to contrast universality versus simplicity. They further show that cotrees are BBF-equivalent to Free(P4) and that Free(P4) is Borel complete but not on top for infinitary bi-interpretability, highlighting a nuanced hierarchy of complexity beyond the finite setting.
Abstract
We study classes of countable graphs where every member does not contain a given finite graph as an induced subgraph -- denoted by $\mathsf{Free}(\mathcal{G})$ for a given finite graph $\mathcal{G}$. Our main results establish a structural dichotomy for such classes: If $\mathcal{G}$ is not an induced subgraph of $\mathcal{P}_4$, then $\mathsf{Free}(\mathcal{G})$ is on top under effective bi-interpretability, implying that the members of $\mathsf{Free}(\mathcal{G})$ exhibit the full range of structural and computational behaviors. In contrast, if $\mathcal{G}$ is an induced subgraph of $\mathcal{P}_4$, then $\mathsf{Free}(\mathcal{G})$ is structurally simple, as witnessed by the fact that every member satisfies the computable embeddability condition. This dichotomy is mirrored in the finite setting when one considers combinatorial and complexity-theoretic properties. Specifically, it is known that $\mathsf{Free}(\mathcal{G})^{fin}$ is complete for graph isomorphism and not a well-quasi-order under embeddability whenever $\mathcal{G}$ is not an induced subgraph of $\mathcal{P}_4$, while in all other cases $\mathsf{Free}(\mathcal{G})^{fin}$ forms a well-quasi-order and the isomorphism problem for $\mathsf{Free}(\mathcal{G})^{fin}$ is solvable in polynomial time.