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.
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