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Toric Mutations in the dP$_2$ Quiver and Subgraphs of the dP$_2$ Brane Tiling

Brane tilings are infinite, bipartite, periodic, planar graphs that are dual to quivers. In this paper, we examine the del Pezzo 2 (dP$_2$) quiver and its brane tiling, which arise from the physics literature, in terms of toric mutations on its corresponding cluster. Specifically, we give explicit formulas for all cluster variables generated by toric mutation sequences. Moreover, for each such variable, we associate a subgraph of the dP$_2$ brane tiling to it such that its weight matches the variable.