  In this article, we give combinatorial formulas for the regularity and the
projective dimension of $3$-path ideals of chordal graphs, extending the
well-known formulas for the edge ideals of chordal graphs given in terms of the
induced matching number and the big height, respectively. As a consequence, we
get that the $3$-path ideal of a chordal graph is Cohen-Macaulay if and only if
it is unmixed. Additionally, we show that the Alexander dual of the $3$-path
ideal of a tree is vertex splittable, thereby resolving the $t=3$ case of a
recent conjecture in [Internat. J. Algebra Comput., 33(3):481--498, 2023].
Also, we give examples of chordal graphs where the duals of their $t$-path
ideals are not vertex splittable for $t\ge 3$. Furthermore, we extend the
formula of the regularity of $3$-path ideals of chordal graphs to all $t$-path
ideals of caterpillar graphs. We then provide some families of graphs to show
that these formulas for the regularity and the projective dimension cannot be
extended to higher $t$-path ideals of chordal graphs (even in the case of
trees).
