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A new parameter-free graph-morphology-based segmentation algorithm is proposed to address the problem of partitioning a 3D triangular mesh into disjoint submeshes that correspond to the physical parts of the underlying object. Curvedness, which is a rotation and translation invariant shape descriptor, is computed at every vertex in the input triangulation. Iterative graph dilation and morphological filtering of the outlier curvedness values result in multiple disjoint maximally connected submeshes such that each submesh contains a set of vertices with similar curvedness values, and vertices in disjoint submeshes have significantly different curvedness values. Experimental evaluations using the triangulations of a number of complex objects demonstrate the robustness and the efficiency of the proposed algorithm and the results prove that it compares well with a number of state-of-the-art mesh segmentation algorithms.