The $t$-connected ideal of a graph $G$ is generated by all connected induced subgraphs of $G$ with $t$ vertices. When $t = 2$, this coincides with the usual edge ideal of the graph. Following the work of Faridi et al., we give a classification of the graphs whose $t$-connected ideals are minimally resolved by their Scarf complex. We also consider the $t$-path ideal of a graph $G$ which is the ideal generated by all paths of length $t$ in $G$. In this case, we are able to give a classification of the same type for paths of length $t = 4$.