On the Chordality of Polynomial Sets in Triangular Decomposition in Top-Down Style

In this paper the chordal graph structures of polynomial sets appearing in triangular decomposition in top-down style are studied when the input polynomial set has a chordal associated graph. We prove that the associated graph of one specific triangular set computed in any algorithm for triangular decomposition in top-down style is a subgraph of the chordal graph of the input polynomial set and that all the polynomial sets, including all the computed triangular sets, appearing in one specific algorithm for triangular decomposition in top-down style (Wang's method) have associated graphs which are subgraphs of the chordal graph of the input polynomial set.