Linear quotients of connected ideals of graphs

As a higher analogue of the edge ideal of a graph, we study the $t$-connected ideal ${\mathrm{J}}_t$. This is the monomial ideal generated by the connected subsets of size $t$. For chordal graphs, we show that ${\mathrm{J}}_t$ 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 ${\mathrm{J}}_t(G)$ has linear quotients and, hence, linear resolution.