[Paper Review] The undecidability of joint embedding and joint homomorphism for hereditary graph classes

Samuel Braunfeld|arXiv (Cornell University)|Mar 28, 2019

semigroups and automata theory4 citations

TL;DR

This paper proves the undecidability of the joint embedding property (JEP) and joint homomorphism property (JHP) for hereditary graph classes defined by finite sets of forbidden induced subgraphs. Using a reduction from the undecidable tiling problem, the author constructs graph classes where JEP and JHP correspond precisely to the existence of a tiling solution, establishing that no algorithm can decide these properties in general.

ABSTRACT

We prove that the joint embedding property is undecidable for hereditary graph classes, via a reduction from the tiling problem. The proof is then adapted to show the undecidability of the joint homomorphism property as well.

Motivation & Objective

• To determine whether the joint embedding property (JEP) is decidable for hereditary graph classes defined by finite sets of forbidden induced subgraphs.
• To investigate the decidability of the joint homomorphism property (JHP) in the same class of graph hereditary classes.
• To extend prior undecidability results on universal countable graphs to the JEP and JHP settings.
• To address open questions on atomicity and universality in permutation classes and monotone graph classes.

Proposed method

• Reduce the undecidable tiling problem to the joint embedding problem by encoding grid and tile structures into graphs with specific induced subgraph constraints.
• Construct two canonical graphs, $A^*$ and $B^*$, representing a grid and a tiling system, respectively, using unary predicates for structural control.
• Impose constraints via forbidden induced subgraphs to ensure that joint embedding of $A^*$ and $B^*$ encodes a valid tiling solution.
• Use the compactness theorem to equate finite JEP with JEP for countable structures, enabling model-theoretic reasoning.
• Adapt the proof to the joint homomorphism property by showing that homomorphisms into a common structure must be embeddings under the constraints.
• Apply coding techniques to translate the enriched language (with unary predicates) into the pure graph language, preserving undecidability.

Figure 1: A portion of the canonical models $A^{*}$ and $B^{*}$ , with the grid points in $A^{*}$ tiled by tiles attached to grid points with the same coordinates in $B^{*}$ . Path points are blue, with the origin a different shade. Grid points are red, their $y$ -coordinate determined by an orange

Experimental results

Research questions

• RQ1 Is there an algorithm that, given a finite set of forbidden induced subgraphs, decides whether the corresponding hereditary graph class has the joint embedding property?
• RQ2 Is there an algorithm that, given a finite set of forbidden induced subgraphs, decides whether the corresponding hereditary graph class has the joint homomorphism property?
• RQ3 Can the methods used for graphs be adapted to show undecidability of atomicity in finitely based permutation classes?
• RQ4 Is the joint embedding property decidable for monotone graph classes (defined by forbidden non-induced subgraphs)?
• RQ5 Can the undecidability of JEP be extended to hereditary classes of permutation graphs defined by forbidden permutations?

Key findings

• The joint embedding property is undecidable for hereditary graph classes defined by finite sets of forbidden induced subgraphs.
• The joint homomorphism property is also undecidable for the same class of graph hereditary classes.
• The undecidability is established via a reduction from the tiling problem, which is known to be undecidable.
• The proof first works in an enriched language with unary predicates and then reduces to the pure graph language using coding techniques.
• The construction ensures that joint embedding of two specific graphs $A^*$ and $B^*$ corresponds exactly to a solution of the tiling problem.
• The result implies that there is no algorithm to decide whether a hereditary graph class admits a universal countable structure.