Polygon mesh is among the most common data structures used for representing objects in computer graphics. Unfortunately, a polygon mesh does not capture high-level structures, unlike a hierarchical model. In general, high-level abstractions are useful for managing data in applications. In this paper, we present a method for decomposing an object represented in polygon meshes into components by means of critical points. The method consists of steps to define the root vertex of the object, define a function on the polygon meshes, compute the geodesic tree and critical points, decide the decomposition order, and extract components using backwards flooding. We have implemented the method. The preliminary results show that it works effectively and efficiently. The decomposition results can be useful for applications such as 3D model retrieval and morphing.