The P vs NP problem asks whether every problem whose solution can be verified in polynomialtime (NP) can also be solved in polynomial time (P). In this paper, we present a proof that P =NP, demonstrating that every NP problem can be solved deterministically in polynomial time usinga graph-based algorithm. We introduce a new Computation Model that enables the simulation ofa Turing machine, and show that NP problems can be simulated efficiently within this framework.By introducing the concept of a Feasible Graph, we ensure that the simulation can be performedin polynomial time, providing a direct path to resolving the P = NP question. Our result hassignificant implications for fields such as cryptography, optimization, and artificial intelligence, whereNP-complete problems play a central role.

This preprint presents a deterministic, graph-based polynomial-time algorithm for NP problems. It has also been submitted to arXiv (https://arxiv.org/pdf/2508.13166). Minor revisions may follow prior to journal submission. DOI establishes priority of this work.