The 2-D architecture of fully programmable valve arrays (FPVAs) is designed as a crossbar consisting of reaction chambers and microvalves, functioning as a versatile, flow-based microfluidic lab-on-chip for implementing biochemical protocols. While an FPVA enables efficient execution of various fluidic operations—such as mixing, loading, and storage, transporting fluids between chambers remains a challenging task. Furthermore, mapping a general mixing tree (representing a sequence of mixing steps) onto an FPVA is complex. It requires careful placement of reagents into specific chambers and the scheduling of subsequent mixing operations. Most sample preparation algorithms aim to generate a minimum-depth mixing tree to achieve the target mixing ratio. However, due to constraints on fluid transportation and scheduling, such a tree may not be the most practical for FPVA implementation. In this article, we harness the power of a satisfiability solver to derive a skewed mixing tree/graph that can be efficiently mapped onto an FPVA using a single mixer. This approach localizes most fluidic operations to a small region of the crossbar. Simulation results show that, for most mixing ratios, a skewed mixing tree can be found which not only reduces fluid-transportation distance and scheduling complexities but also the number of loading cycles, reagent volumes, and waste production in sample preparation, when compared to the approach based on the minimum-depth mixing tree.