The literature on the watershed is separated in two families: the watersheds on node weighted graphs and the watersheds on edge weighted graphs. The simplest node weighted graphs are images, where the nodes are the pixels ; neighboring pixels being linked by unweighted pixels. The edge weights on an edge weighted graph express dissimilarities between the unweighted nodes. Distinct definitions of minima and catchment basins have been given for both types of graphs from which different algorithms have been derived. This paper aims at showing that watersheds on edge or node weighted graphs are strictly equivalent. Moreover, all algorithms developed for edge weighted graphs may be applied on node weighted graphs and vice versa. From any node or edge weighted graph it is possible to derive a flooding graph with node and edge weights. Its regional minima and catchment basins are identical whether one considers the node weights alone or the edge weights alone. A lexicographic order relation permits to compare non ascending paths with the same origin according to their steepness. Overlapping zones between neighboring catchment basins are reduced or even suppressed by pruning edges in the flooding graph through which does not pass a steepest path and reduces, without arbitrary choices the overlapping zones between adjacent catchment basins. We propose several ways to break the remaining ties, the simplest being to assign slightly distinct weights to regional minima with the same weight. Like that each node is linked with only one regional minimum by a path of maximal steepness.
