Register allocation is often a two-phase approach: spilling of registers to memory, followed by coalescing of registers. Extreme live-range splitting (\ie live-range splitting after each statement) enables optimal solutions based on ILP, for both spilling and coalescing. However, while the solutions are easily found for spilling, for coalescing they are more elusive. This difficulty stems from the huge size of interference graphs resulting from live-range splitting. This report focuses on optimal coalescing in the context of extreme live-range splitting. We present some theoretical properties that give rise to an algorithm for reducing interference graphs, while preserving optimality. This reduction consists mainly in finding and removing useless splitting points. It is followed by a graph decomposition based on clique separators. The last optimization consists in two preprocessing rules. Any coalescing technique can be applied after these optimizations. Our optimizations have been tested on a standard benchmark, the optimal coalescing challenge. For this benchmark, the cutting-plane algorithm for optimal coalescing (the only optimal algorithm for coalescing) runs 300 times faster when combined with our optimizations. Moreover, we provide all the solutions of the optimal coalescing challenge, including the 3 instances that were previously unsolved.