A $n$-item list can be verified as sorted by comparing every item to its neighbor. In my application, I will not be able to compare every item with its neighbor: instead, the comparisons will sometimes be between distant elements. Given that the list contains more than three items and also that comparison is the only supported operation, does there ever exist a "network" of comparisons that will prove that the list is sorted, but is missing at least one direct neighbor-to-neighbor comparison?
Formally, for a sequence of elements $e_i$, I have a set of pairs of indices $(j,k)$ for which I know whether $e_j > e_k$, $e_j = e_k$, or $e_j < e_k$. There exists a pair $(l,l+1)$ that is missing from the set of comparisons. Is it ever possible, then, to prove that the sequence is sorted?