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A common problem shared by several leading morphological shape representation algorithms is that there is much overlapping among the representative disks of the same size. A shape component represented by a group of connected disk centers sometimes uses many heavily overlapping representative disks to represent a relatively simple shape part. A shape component may also contain a large number of representative disks that form a complicated structure. We introduce a generalized discrete morphological skeleton transform that uses eight structuring elements to generate skeleton subsets so that no two skeletal points from the same skeleton subset are adjacent to each other. Each skeletal point represents a shape part that is in general an octagon with four pairs of parallel opposing sides. The number of representative points needed to represent a given shape is significantly lower than that in the standard skeleton transform. A collection of shape components needed to build a structural representation is easily derived from the generalized skeleton transform. Each shape component covers a significant area of the given shape and severe overlapping is avoided. The given shape can also be accurately approximated using a small number of shape components.