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The Konishi anomalies for noncommutative N=1 supersymmetric U(1) gauge theoryarising from planar and nonplanar diagrams are calculated. Whereas planarKonishi anomaly is the expected \star-deformation of the commutative anomaly,nonplanar anomaly reflects the important features of nonplanar diagrams ofnoncommutative gauge theories, such as UV/IR mixing and the appearance ofnonlocal open Wilson lines. We use the planar and nonplanar Konishi anomaliesto calculate the effective superpotential of the theory. In the limit ofvanishing |\Theta p|, with \Theta the noncommutativity parameter, thenoncommutative effective superpotential depends on a gauge invariantsuperfield, which includes supersymmetric Wilson lines, and has nontrivialdependence on the gauge field supermultiplet.Comment: LaTeX, 36 pages. Version 2: Typos Corrected. Version 3: Extensively  revised version, 42 pages, to be published in Int. J. Mod. Phys. A. (2005

PLANAR AND NONPLANAR KONISHI ANOMALIES AND EFFECTIVE SUPERPOTENTIAL FOR NONCOMMUTATIVE ${\mathcal N}=1$ SUPERSYMMETRIC U(1)