  We study the SHORTEST PATH problem with positive disjunctive constraints from
the perspective of parameterized complexity. For positive disjunctive
constraints, there are certain pair of edges such that any feasible solution
must contain at least one edge from every such pair. In this paper, we initiate
the study of SHORTEST PATH problem subject to some positive disjunctive
constraints the classical version is known to be NP-Complete. Formally, given
an undirected graph G = (V, E) with a forcing graph H = (E, F) such that the
vertex set of H is same as the edge set of G. The goal is to find a set S of at
most k edges from G such that S forms a vertex cover in H and there is a path
from s to t in the subgraph of G induced by the edge set S. In this paper, we
consider two natural parameterizations for this problem. One natural parameter
is the solution size, i.e. k for which we provide a kernel with O(k^5) vertices
when both G and H are general graphs. Additionally, when either G or H (but not
both) belongs to some special graph classes, we provied kernelization results
with O(k^3) vertices . The other natural parameter we consider is structural
properties of H, i.e. the size of a vertex deletion set of H to some special
graph classes. We provide some fixed-parameter tractability results for those
structural parameterizations.
