Reachability in Dynamic Line Graph
Nomenclature aside (a line graph is something different than what you describe), you appear to be describing the problem of ''maintaining order in a list''. You have a list of items $v_i$ (your graph), you can remove an item from one point in the list and re-insert it elsewhere, and you want to check whether one item appears earlier than another in the list. This problem has a constant-time solution; see Dietz and Sleator, "Two algorithms for maintaining order in a list", STOC 1987, https://www.cs.cmu.edu/~sleator/papers/maintaining-order.html and https://en.wikipedia.org/wiki/Order-maintenance_problem