Let $S$ be a string over some alphabet $\Sigma$. It is well known that a substring of $S$ is commonly defined as a sequence of contiguous elements from $S$, while a subsequence of $S$ is a sequence made of non necessarily contiguous elements from $S$ (e.g. if $S="123465835"$, then $"4658"$ is a substring of $S$ while $"1236"$ is a subsequence of $S$). But is there a word to refer to subsequences that can be obtained from $S$ by removing an arbitrary substring (e.g. $"12835"$)?


(this concept seems complementary to that of a subsequence, hence the conjectured "co-subsequence" in the title -- although "co-substring" might be a good choice too)



Strings and "co-subsequences"
Since you are taking the complement of a substring, and it appears that there may be no firmly established terminology, I propose:


* a *substring complement* is what remains after deleting a substring,
* and more generally, a *subsequence complement* is what remains after deleting a subsequence.


Thus, one may refer to the substring complement of s in t, and use the notation t - s, or $t \setminus s$, with the same notation for the subsequence.


I would prefer this natural language terminology over the alternative co-substring and co-subsequence, which sound unnecessarily technical to my ear, but this difference may be slight. 


It does seem worthwhile, however, to distinguish between the two cases, and so I would argue against using the term co-subsequence, as you suggested in your question, to refer to the substring complement.


