Constructing self-referential instances for the clique problem

In this paper, we propose constructing self-referential instances to reveal the inherent algorithmic hardness of the clique problem. First, we prove the existence of a phase transition phenomenon for the clique problem in the Erd\H{o}s--R\'enyi random graph model and derive an exact location for the transition point. Subsequently, at the transition point, we construct a family of graphs. In this family, each graph shares the same number of vertices, number of edges, and degree sequence, yet both instances containing a $k$-clique and instances without any $k$-clique are included. These two states can be transformed into each other through a symmetric transformation that preserves the degree of every vertex. This property explains why exhaustive search is required in the critical region: an algorithm must search nearly the entire solution space to determine the existence of a solution; otherwise, a counterinstance can be constructed from the original instance using the symmetric transformation. Finally, this paper elaborates on the intrinsic reason for this phenomenon from the independence of the solution space.