Dynamic programming on bounded treewidth
The reason a small width turns intractable problems linear.
The pattern
Given a tree decomposition of width k, a vast class of problems — maximum independent set, dominating set, graph coloring, Hamiltonicity — can be solved by dynamic programming over the bags. You process the tree from leaves to root, and at each bag you maintain a table indexed by the possible configurations of the at-most-(k+1) vertices in that bag.
Because each bag has at most k+1 vertices, each table has at most a constant (in n) number of rows — exponential in k but independent of the graph size. The total running time is f(k)·n: exponential in the width, linear in the graph. That is the whole payoff of small treewidth.
Why the connectivity condition pays off here
When you move from a child bag to its parent, vertices that appear in the child but not the parent are forgotten, and the connectivity condition guarantees they will never reappear. So you can project them out of the table immediately, keeping it small. Without that guarantee the table would grow as you sweep, and the linear-time promise would collapse.
Questions
What does f(k)·n mean?
The running time is some function of the treewidth k (often 2^O(k) or k^O(k)) multiplied by the number of vertices n. Fix k and it is linear in n.
Does this work for every graph problem?
For every problem expressible in monadic second-order logic, yes — that is Courcelle's theorem. Many problems outside that class also yield to it.