  In this paper, we first prove that when the associated graph of a polynomial
set is chordal, a particular triangular set computed by a general algorithm in
top-down style for computing the triangular decomposition of this polynomial
set has an associated graph as a subgraph of this chordal graph. Then for
Wang's method and a subresultant-based algorithm for triangular decomposition
in top-down style and for a subresultant-based algorithm for regular
decomposition in top-down style, we prove that all the polynomial sets
appearing in the process of triangular decomposition with any of these
algorithms have associated graphs as subgraphs of this chordal graph. These
theoretical results can be viewed as non-trivial polynomial generalization of
existing ones for sparse Gaussian elimination, inspired by which we further
propose an algorithm for sparse triangular decomposition in top-down style by
making use of the chordal structure of the polynomial set. The effectiveness of
the proposed algorithm for triangular decomposition, when the polynomial set is
chordal and sparse with respect to the variables, is demonstrated by
preliminary experimental results.
