Matchings and Path Covers with applications to Domination in Graphs

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, $α'(G)$, 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 $α'(G) + \frac{1}{2}\rm{pc}(G) \ge \frac{n}{2}$ and we provide a constructive characterization of the graphs achieving equality in this bound. It is known that $γ(G) \le α'(G)$ and $γ_t(G) \le α'(G) + \rm{pc}(G)$, where $γ(G)$ and $γ_t(G)$ 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 $δ(G) \ge 3$, then $γ_t(G) \le α'(G) + \frac{1}{2}(\rm{pc}(G) - 1)$, 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, $γ_{\rm nt}(G)$, is the minimum cardinality of a neighborhood total dominating set of $G$. We observe that $γ(G) \le γ_{\rm nt}(G) \le γ_t(G)$. 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 $γ_{\rm nt}(G) \le α'(G) + \frac{1}{2}\rm{pc}(G)$ and this bound is tight.

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