We present three equivalent models for graphs and their covers, two of which are applicable to more general structures such as mixed colored directed multigraphs with semi-edges. In this context, we extend the concept of equitable and degree partitions, and provide efficient algorithms for their calculation. We demonstrate that the dart model introduces simpler concepts. Leveraging this model, we introduce a novel notion of the degree matrix for the general graph model. Additionally, we reassert and expand upon Leighton's theorem regarding the existence of a finite common cover for graphs containing semi-edges. [ABSTRACT FROM AUTHOR]

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