The k-syntactic similarity approach is couched in graphical representation terms and its ability to provide global recognition capability while retaining a low time complexity is explored. One potential application domain, that of composite shape decomposition into approximately convex subshapes, is described. This is shown to be equivalent to finding cycles within a particular graph. The approach yields valid decompositions in many cases of interest, and is capable of identifying those cases where additional semantic considerations are necessary for proper analysis. The permissible graph structures representing composite shapes given a reasonable set of relations are determined. Experimental results on nonideal data are given.