Connectivity Preserving Iterative Compaction and Finding 2 Disjoint Rooted Paths in Linear Time

In this paper we show how to combine two algorithmic techniques to obtain linear time algorithms for various optimization problems on graphs, and present a subroutine which will be useful in doing so. The first technique is iterative shrinking. In the first phase of an iterative shrinking algorithm, we construct a sequence of graphs of decreasing size G1, . . . , Gl where G1 is the initial input, Gl is a graph on which the problem is easy(often because it is small), and Gi is obtained from Gi+1 via some shrinking algorithm. In the second phase we work through the sequence in reverse, repeatedly constructing a solution for a graph from the solution for its successor. In an iterative compaction algorithm, we insist that the shrinking algorithm is actually a compaction algorithm, i.e. for some constant δ > 0, for every consecutive pair, Gi, Gi+1 of the sequence we have |V (Gi+1)|+ |E(Gi+1| ≤ (1− δ)(|V (Gi)|+ |E(Gi)|). Another approach to solving optimization problems is to exploit the structural properties implied by the connectivity of the input graph. Thus, both isomorphism testing and planar embedding are easier for 3-connected planar graphs because these graphs have a unique embedding. This approach can be used on graphs which are not highly connected by decomposing an input graph into its highly connected pieces, solving subproblems on these specially structured pieces and then combining their solutions. This usually involves building, explicitly or implicitly, a tree decomposition of bounded adhesion and working with the pieces into which it splits the input. We combine these two techniques by developing compaction algorithms which when applied to the highly connected pieces preserve their connectivity properties. The structural properties this connectivity implies can be helpful both in finding further compactions in later iterations and when we are manipulating solutions in the second phase of an iterative compaction algorithm which uses such a compaction