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This paper deals with the mathematical properties of watersheds in weighted graphs linked to region merging methods, as used in image analysis. In a graph, a cleft (or a binary watershed) is a set of vertices that cannot be reduced, by point removal, without changing the number of regions (connected components) of its complement. To obtain a watershed adapted to morphological region merging, it has been shown that one has to use the topological thinnings introduced by M. Couprie and G. Bertrand. Unfortunately, topological thinnings do not always produce thin clefts. Therefore, we introduce a new transformation on vertex weighted graphs, called C-watershed, that always produces a cleft. We present the class of perfect fusion graphs, for which any two neighboring regions can be merged, while preserving all other regions, by removing from the cleft the points adjacent to both. An important theorem of this paper states that, on these graphs, the C-watersheds are topological thinnings and the corresponding divides are thin clefts. We propose a linear-time immersion-like algorithm to compute C-watersheds on perfect fusion graphs, whereas, in general, a linear-time topological thinning algorithm does not exist. Furthermore, we prove that this algorithm is monotone in the sense that the vertices are processed in increasing order of weight. Finally, we derive some characterizations of perfect fusion graphs based on the thinness properties of both C-watersheds and topological watersheds.