  Let $G$ be a graph with no isolated vertex. A matching in $G$ is a set of
edges that are pairwise not adjacent in $G$, while the matching number,
$\alpha'(G)$, of $G$ is the maximum size of a matching in $G$. The path
covering number, $\rm{pc}(G)$, of $G$ is the minimum number of vertex disjoint
paths such that every vertex belongs to a path in the cover. We show that if
$G$ has order $n$, then $\alpha'(G) + \frac{1}{2}\rm{pc}(G) \ge \frac{n}{2}$
and we provide a constructive characterization of the graphs achieving equality
in this bound. It is known that $\gamma(G) \le \alpha'(G)$ and $\gamma_t(G) \le
\alpha'(G) + \rm{pc}(G)$, where $\gamma(G)$ and $\gamma_t(G)$ denote the
domination and the total domination number of $G$. As an application of our
result on the matching and path cover numbers, we show that if $G$ is a graph
with $\delta(G) \ge 3$, then $\gamma_t(G) \le \alpha'(G) +
\frac{1}{2}(\rm{pc}(G) - 1)$, and this bound is tight. A set $S$ of vertices in
$G$ is a neighborhood total dominating set of $G$ if it is a dominating set of
$G$ with the property that the subgraph induced by the open neighborhood of the
set $S$ has no isolated vertex. The neighborhood total domination number,
$\gamma_{\rm nt}(G)$, is the minimum cardinality of a neighborhood total
dominating set of $G$. We observe that $\gamma(G) \le \gamma_{\rm nt}(G) \le
\gamma_t(G)$. As a further application of our result on the matching and path
cover numbers, we show that if $G$ is a connected graph on at least six
vertices, then $\gamma_{\rm nt}(G) \le \alpha'(G) + \frac{1}{2}\rm{pc}(G)$ and
this bound is tight.
