On the joint embedding property for cographs and trees

TL;DR

This paper investigates the joint embedding property (JEP) for specific graph families, proving decidability in cases involving cographs, trees, and bounded treewidth or cliquewidth families, with implications for graph theory and computational complexity.

Contribution

It establishes decidability of the JEP for certain classes of graphs and tree families, extending previous undecidability results.

Findings

Decidability of JEP when P4 is in the forbidden set.

Generalization to rooted labeled trees under topological containment.

Applicability to bounded treewidth and cliquewidth families.

Abstract

A family of graphs $\mathcal{F}$ is said to have the joint embedding property (JEP) if for every $G_1, G_2\in \mathcal{F}$, there is an $H\in \mathcal{F}$ that contains both $G_1$ and $G_2$ as induced subgraphs. If $\mathcal{F}$ is given by a finite set $S$ of forbidden induced subgraphs, it is known that determining if $\mathcal{F}$ has JEP is undecidable. We prove that this problem is decidable if $P_4\in S$ and generalize this result to families of rooted labeled trees under topological containment, bounded treewidth families under the graph minor relation, and bounded cliquewidth families under the induced subgraph relation.