  We consider a variant of the path cover problem, namely, the
$k$-fixed-endpoint path cover problem, or kPC for short, on interval graphs.
Given a graph $G$ and a subset $\mathcal{T}$ of $k$ vertices of $V(G)$, a
$k$-fixed-endpoint path cover of $G$ with respect to $\mathcal{T}$ is a set of
vertex-disjoint paths $\mathcal{P}$ that covers the vertices of $G$ such that
the $k$ vertices of $\mathcal{T}$ are all endpoints of the paths in
$\mathcal{P}$. The kPC problem is to find a $k$-fixed-endpoint path cover of
$G$ of minimum cardinality; note that, if $\mathcal{T}$ is empty the stated
problem coincides with the classical path cover problem. In this paper, we
study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for
short, generalizing the 1HP problem which has been proved to be NP-complete
even for small classes of graphs. Motivated by a work of Damaschke, where he
left both 1HP and 2HP problems open for the class of interval graphs, we show
that the 1PC problem can be solved in polynomial time on the class of interval
graphs. The proposed algorithm is simple, runs in $O(n^2)$ time, requires
linear space, and also enables us to solve the 1HP problem on interval graphs
within the same time and space complexity.
