We present a new polynomial-time algorithm for determining whether two given graphs are isomorphic or not. We prove that the algorithm is necessary and sufficient for solving the Graph Isomorphism Problem in polynomial-time, thus showing that the Graph Isomorphism Problem is in P. The semiotic theory for the recognition of graph structure is used to define a canonical form of the sign matrix of a graph. We prove that the canonical form of the sign matrix is uniquely identifiable in polynomial-time for isomorphic graphs. The algorithm is demonstrated by solving the Graph Isomorphism Problem for many of the hardest known examples. We implement the algorithm in C++ and provide a demonstration program for Microsoft Windows.