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.

翻译：我们研究了Erdos-Renyi随机图模型$G(n,p)$中支配数的算法可判定性。通过精心选择边概率$p=p(n)$，我们证明支配问题展现出强烈的不可约性质。具体而言，对于任意常数$0<c<1$，任何仅检查规模不超过$n^c$的导出子图的算法，都无法判定$G(n,p)$是否包含规模为$k=\ln n$的支配集。我们通过构造仅改变常数条边的局部对称映射，可以翻转此类支配集的存在性，从而生成需要穷举搜索才能区分的随机图实例。这些结果表明，随机图中支配集问题的极端困难性不能归因于局部结构，而是源于整个解空间的自指特性和近独立结构。