Self-referential instances of the dominating set problem are irreducible

TL;DR

This paper proves that determining the domination number in certain random graphs is highly complex, with local algorithms failing due to the problem's self-referential and global structural properties.

Contribution

It introduces a new irreducibility property of the domination problem in Erdős-Rényi graphs, showing local inspection cannot decide the existence of small dominating sets.

Findings

No small induced subgraph inspection can determine the domination number.

Existence of dominating sets can be flipped by local symmetry mappings.

Hardness arises from global, self-referential graph properties, not local structure.

Abstract

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 $n^c$ 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.