Matchings, path covers and domination

We show that if G is a graph with minimum degree at least three, then γ _t ( G ) ≤ α ^′ ( G ) + ( pc ( G ) − 1 ) ∕ 2 and this bound is tight, where γ _t ( G ) is the total domination number of G, α ^′ ( G ) the matching number of G and pc ( G ) the path covering number of G which is the minimum number of vertex disjoint paths such that every vertex belongs to a path in the cover. We show that if G is a connected graph on at least six vertices, then γ _nt ( G ) ≤ α ^′ ( G ) + pc ( G ) ∕ 2 and this bound is tight, where γ _nt ( G ) denotes the neighborhood total domination number of G. We observe that every graph G of order n satisfies α ^′ ( G ) + pc ( G ) ∕ 2 ≥ n ∕ 2, and we characterize the trees achieving equality in this bound.