Matchings and Path Covers with applications to Domination in Graphs

Abstract

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.