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On the joint embedding property for cographs and trees

Daniel Carter

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.

On the joint embedding property for cographs and trees

Abstract

A family of graphs is said to have the joint embedding property (JEP) if for every , there is an that contains both and as induced subgraphs. If is given by a finite set of forbidden induced subgraphs, it is known that determining if has JEP is undecidable. We prove that this problem is decidable if 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.
Paper Structure (11 sections, 12 theorems, 5 equations)

This paper contains 11 sections, 12 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. JEP for tree languages
  3. Explicitly and implicitly regular families
  4. A more explicit construction
  5. Applications
  6. General trees
  7. Cographs
  8. Bounded treewidth families
  9. Bounded cliquewidth families
  10. Further discussion
  11. Acknowledgements

Key Result

theorem 1.1

Problem prob:treejep is decidable. In fact, it is decidable to determine if $(L,\preceq)$ has JEP for any regular language of trees $L$, if a tree automaton recognizing $L$ is given as input.

Theorems & Definitions (24)

  • theorem 1.1
  • corollary 1
  • corollary 2
  • corollary 3
  • corollary 4
  • proposition 1
  • proof
  • proposition 2
  • lemma 1
  • lemma 2
  • ...and 14 more