  Given a graph $G = (V,E)$, a vertex $u \in V$ ve-dominates all edges incident
to any vertex of $N_G[u]$. A set $S \subseteq V$ is a ve-dominating set if for
all edges $e\in E$, there exists a vertex $u \in S$ such that $u$ ve-dominates
$e$. Lewis [Ph.D. thesis, 2007] proposed a linear time algorithm for
ve-domination problem for trees. In this paper, first we have constructed an
example where the proposed algorithm fails. Then we have proposed a linear time
algorithm for ve-domination problem in block graphs, which is a superclass of
trees. We have also proved that finding minimum ve-dominating set is
NP-complete for undirected path graphs. Finally, we have characterized the
trees with equal ve-domination and independent ve-domination number.
