A note on the open packing number in graphs
Mehdi Mohammadi, Mohammad Maghasedi · Mathematica Bohemica · 2018
A subset $S$ of vertices in a graph $G$ is an open packing set if no pair of vertices of $S$ has a common neighbor in $G$. An open packing set which is not a proper subset of any open packing set is called a maximal open packing set. The maximum cardinality of an open packing set is called the open packing number and is denoted by $\rho^{\rm o}(G)$. A subset $S$ in a graph $G$ with no isolated vertex is called a total dominating set if any vertex of $G$ is adjacent to some vertex of $S$. The total domination number of $G$, denoted by $\gamma_t(G)$, is the minimum cardinality of a total dominating set of $G$. We characterize graphs of order $n$ and minimium degree at least two with $\rho^{\rm o}(G)=\gamma_t(G)=\frac12n$.