On the Computational Complexity of Length- and Neighborhood-Constrained Path Problems

Finding paths in graphs is a fundamental graph-theoretic task. In this work, we we are concerned with finding a path with some constraints on its length and the number of vertices neighboring the path, that is, being outside of and incident with the path. Herein, we consider short and long paths on the one side, and small and large neighborhoods on the other side---yielding four decision problems. We show that all four problems are NP-complete, even in planar graphs with small maximum degree. Moreover, we study all four variants when parameterized by a bound $k$ on the length of the path, by a bound $\ell$ on the size of neighborhood, and by $k + \ell$.