While it is well-known and acknowledged that the performance of graph algorithms is heavily dependent on the input data, there has been surprisingly little research to quantify and predict the impact the graph structure has on performance. Parallel graph algorithms, running on many-core systems such as GPUs, are no exception: most research has focused on how to efficiently implement and tune different graph operations on a specific GPU. However, the performance impact of the input graph has only been taken into account indirectly as a result of the graphs used to benchmark the system. In this work, we present a case study investigating how to use the properties of the input graph to improve the performance of the breadth-first search (BFS) graph traversal. To do so, we first study the performance variation of 15 different BFS implementations across 248 graphs. Using this performance data, we show that significant speed-up can be achieved by combining the best implementation for each level of the traversal. To make use of this data-dependent optimization, we must correctly predict the relative performance of algorithms per graph level, and enable dynamic switching to the optimal algorithm for each level at runtime. We use the collected performance data to train a binary decision tree, to enable high-accuracy predictions and fast switching. We demonstrate empirically that our decision tree is both fast enough to allow dynamic switching between implementations, without noticeable overhead, and accurate enough in its prediction to enable significant BFS speedup. We conclude that our model-driven approach (1) enables BFS to outperform state of the art GPU algorithms, and (2) can be adapted for other BFS variants, other algorithms, or more specific datasets.