Abstract : Register allocation is often a two-phase approach: spilling of registers to memory, followed by coalescing of registers. Extreme liverange splitting (i.e. 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 paper focuses on coalescing in the context of extreme liverange splitting. It presents some theoretical properties that give rise to an algorithm for reducing interference graphs. This reduction consists mainly in finding and removing useless splitting points. It is followed by a graph decomposition based on clique separators. The reduction and decomposition are general enough, so that any coalescing algorithm can be applied afterwards. Our strategy for reducing and decomposing interference graphs preserves the optimality of coalescing. When used together with an optimal coalescing algorithm (e.g. ILP), optimal solutions are much more easily found. The strategy has 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 strategy. Moreover, we provide all the optimal solutions of the optimal coalescing challenge, including the three instances that were previously unsolved.