Minimum bandwidth problem is to a find an ordering of graph nodes on integer line that minimizes the largest distance between any two adjacent nodes. A $k$-caterpillar is a tree formed from main path by growing edge-disjoint paths of length at most $k$ from its nodes ($k$ is called the hair length). Minimum Bandwidth problem is in $P$ for 2-caterpillars but it is $NP$-complete for 3-caterpillars.
Here is a very interesting fact, Minimum bandwidth problem is solvable in polynomial time for 1-caterpillars (hair length at most one) but it is $NP$-complete for cyclic 1-caterpillars (in cyclic caterpillar, one edge is added to connect the endpoints of the main path). So, the addition of exactly one edge makes the problem $NP$-complete.
What is the most striking example of problem hardness jump where a small variation of input instance causes a complexity jump from polynomial-time solvability to $NP$-completeness?