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The edge-reduction of a simple regular graph is an operation which removes two vertices and preserves the regularity. It has played an important role in the study of cubic graphs
[6,7,8]. Our main purpose is to study the structure of edge-irreducible quartic graphs. All edge-irreducible quartic graphs are determined from a constructive viewpoint. Then, a
unique decomposition theorem for edge-irreducible quartic graphs is obtained.