文摘
We show that if G is a graph with minimum degree at least three, then e6f49bc831db7fa1a8b8ce3" title="Click to view the MathML source">γt(G)≤α′(G)+(pc(G)−1)∕2 and this bound is tight, where b02abf428c7d5fb13d981" title="Click to view the MathML source">γt(G) is the total domination number of G, e55084169e5df4e12290d1beb74d1b7" title="Click to view the MathML source">α′(G) the matching number of G and e6b2ddf75c55396e75dedbc5" title="Click to view the MathML source">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 γnt(G)≤α′(G)+pc(G)∕2 and this bound is tight, where 9ca911e0eaf83fc62a3c" title="Click to view the MathML source">γnt(G) denotes the neighborhood total domination number of G. We observe that every graph G of order 9fd938d18098980da0e66388b7" title="Click to view the MathML source">n satisfies 9f5" title="Click to view the MathML source">α′(G)+pc(G)∕2≥n∕2, and we characterize the trees achieving equality in this bound.