On the joint embedding property for cographs and trees

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.

14 pages