Hierarchical image segmentation provides a region-oriented scale-space, {\em i.e.}, a set of image segmentations at different detail levels in which the segmentations at finer levels are nested with respect to those at coarser levels. However, most image segmentation algorithms, among which a graph-based image segmentation method relying on a region merging criterion was proposed by Felzenszwalb-Huttenlocher in 2004, do not lead to a hierarchy. In order to cope with a demand for hierarchical segmentation, Guimar\~aes {\em et al.} proposed in 2012 a method for hierarchizing the popular Felzenszwalb-Huttenlocher method, without providing an algorithm to compute the proposed hierarchy. This article is devoted to provide a series of algorithms to compute the result of this hierarchical graph-based image segmentation method efficiently, based mainly on two ideas: optimal dissimilarity measuring and incremental update of the hierarchical structure. Experiments show that, for an image of size 321 $\times$ 481 pixels, the most efficient algorithm produces the result in half a second whereas the most naive one requires more than four hours.