  As a higher analogue of the edge ideal of a graph, we study the $t$-connected
ideal $\operatorname{J}_{t}$. This is the monomial ideal generated by the
connected subsets of size $t$. For trees, we show that $\operatorname{J}_{t}$
has a linear resolution iff the tree is $t$-gap-free, and that this is
equivalent to having linear quotients. We then show that if $G$ is any gap-free
and $t$-claw-free graph, then $\operatorname{J}_{t}(G)$ has linear quotients
and hence, linear resolution.
