Path Contraction Faster Than 2n
This paper addresses the Path Contraction problem: given a graph $G$, determine whether it can be transformed into a path graph $P_k$ of length $k$ via a sequence of edge contractions. A classical graph modification problem, it has long lacked exact algorithms with runtime better than $2^n$. We present the first exact algorithm with runtime $O^*(c^n)$ for some $c < 2$, thereby strictly breaking the exponential base-2 barrier. Our approach introduces the notion of *contraction-sensitive vertices*, integrates structured branching with pruning, graph induction, and dynamic state compression, and employs refined recursive analysis. The theoretical correctness is fully established. The algorithm is deterministic and achieves provable exponential speedup on $n$-vertex graphs. Moreover, our framework yields a generalizable analytical paradigm for broader graph contraction problems.