  For a given clutter $\mathcal{C}$, let $I:=I ( \bar{\mathcal{C}} )$ be the
circuit ideal in the polynomial ring $S$. In this paper, we show that the Betti
numbers of $I$ and $I + ( \textbf{x}_F )$ are the same in their non-linear
strands, for some suitable $F \in \mathcal{C}$. Motivated by this result, we
introduce a class of clutters that we call chordal. This class, is a natural
extension of the class of chordal graphs and has the nice property that the
circuit ideal associated to any member of this class has a linear resolution
over any field. Finally we compare this class with all known families of
clutters which generalize the notion of chordality, and show that our class
contains several important previously defined classes of chordal clutters. We
also show that in comparison with others, this class is possibly the best
approximation to the class of $d$-uniform clutters with linear resolution over
any field.
