This paper presents a new method for finding complete information about the set of all nonnegative integer solutions of homogeneous and inhomogeneous linear diophantine equations. Such solutions are fundamental for associative-commutative unification. Our algorithm finds all minimal solutions as “monotone” paths in a graph which encodes the linear diophantine equation. This encoding makes repeated arithmetic operations obsolete and allows inexpensive tests for minimality of solutions. This graph algorithm compares favourably with the known methods, namely lexicogragraphic algorithm and completion procedure. A PASCAL implementation can be found in the Appendix.