  Path cover is a well-known intractable problem that finds a minimum number of
vertex disjoint paths in a given graph to cover all the vertices. We show that
a variant, where the objective function is not the number of paths but the
number of length-$0$ paths (that is, isolated vertices), turns out to be
polynomial-time solvable. We further show that another variant, where the
objective function is the total number of length-$0$ and length-$1$ paths, is
also polynomial-time solvable. Both variants find applications in approximating
the two-machine flow-shop scheduling problem in which job processing has
constraints that are formulated as a conflict graph. For the unit jobs, we
present a $4/3$-approximation algorithm for the scheduling problem with an
arbitrary conflict graph, based on the exact algorithm for the variants of the
path cover problem. For the arbitrary jobs while the conflict graph is the
union of two disjoint cliques, that is, all the jobs can be partitioned into
two groups such that the jobs in a group are pairwise conflicting, we present a
simple $3/2$-approximation algorithm.
