A Polynomial Time Algorithm For Solving Clique Problems

I present a single algorithm which solves the clique problems, “What is the largest size clique?”, “What are all the maximal cliques?” and the decision problem, “Does a clique of size k exist?” for any given graph in polynomial time. The existence of this algorithm proves that P = NP.

💡 Research Summary

The manuscript claims to have discovered a single polynomial‑time algorithm that solves all three classic formulations of the clique problem: (i) determining the size of the largest clique, (ii) enumerating all maximal cliques, and (iii) deciding whether a clique of a given size k exists. The author presents the algorithm in two phases, provides illustrative examples on small graphs, and asserts that the method proves P = NP.

Phase 1 – Neighbor Introductions
Each vertex broadcasts its adjacency list to all of its immediate neighbors. Upon receiving a neighbor’s list, a vertex checks whether any of the advertised vertices are also its own neighbors. If a vertex u discovers that both v and w are mutual neighbors (i.e., edges (u,v), (u,w), and (v,w) all exist), it records the triple