# Self-referential instances of the dominating set problem are irreducible

arXiv: https://arxiv.org/abs/2602.10559 · HTML: https://mlsanity.com/paper/2602.10559/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Feb 2026.

## Built on by (most cited first, 1 of 1)

- [Can Machines Really See Objects in Images? A Study Based on Syntactic Distance and Visual Self-Referential Instances](https://mlsanity.com/paper/2606.29416/) (2026)

## Abstract (arXiv)

We study the algorithmic decidability of the domination number in the Erdos-Renyi random graph model G(n,p). We show that for a carefully chosen edge probability p=p(n), the domination problem exhibits a strong irreducible property. Specifically, for any constant 0<c<1, no algorithm that inspects only an induced subgraph of order at most nc can determine whether G(n,p) contains a dominating set of size k=ln n. We demonstrate that the existence of such a dominating set can be flipped by a local symmetry mapping that alters only a constant number of edges, thereby producing indistinguishable random graph instances which require exhaustive search. These results demonstrate that the extreme hardness of the dominating set problem in random graphs cannot be attributed to local structure, but instead arises from the self-referential nature and near-independence structure of the entire solution space.