Matchings and Path Covers with applications to Domination in Graphs

TL;DR

This paper establishes a new inequality relating matching number and path covering number in graphs, characterizes extremal graphs, and applies these results to bounds on various domination parameters.

Contribution

It provides a novel bound connecting matching and path cover numbers and applies it to improve bounds on domination and total domination numbers in graphs.

Findings

Proves that '(G) + rac{1}{2} m{pc}(G) \u2265 rac{n}{2} for graphs with no isolated vertices.

Characterizes graphs achieving equality in the main bound.

Derives tight bounds on total domination and neighborhood total domination numbers using the main result.

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.