Dichotomy results for classes of countable graphs

TL;DR

This paper establishes a structural dichotomy for classes of countable graphs defined by forbidden induced subgraphs, showing a sharp contrast in complexity and structural properties depending on the subgraph's relation to a path of four vertices.

Contribution

It proves a dichotomy theorem classifying classes of countable graphs based on forbidden subgraphs, linking structural simplicity with computational tractability.

Findings

If the forbidden subgraph is not an induced subgraph of P4, the class exhibits full structural and computational complexity.

If the forbidden subgraph is an induced subgraph of P4, the class is structurally simple with computable embeddability.

Finite versions of these classes are either graph isomorphism complete or well-quasi-ordered, with polynomial-time isomorphism algorithms.

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.