Geodesic
A note on the open packing number in graphs
Mathematica Bohemica, Tome 144 (2019) no. 2, pp. 221-224 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\rho ^{\rm o}(G)=\gamma _t(G)=\frac 12n$.
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 $\rho ^{\rm o}(G)=\gamma _t(G)=\frac 12n$.
DOI : 10.21136/MB.2018.0124-17
Classification : 05C69, 05C70
Keywords: packing; open packing; total domination 
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Mohammadi, Mehdi; Maghasedi, Mohammad. A note on the open packing number in graphs. Mathematica Bohemica, Tome 144 (2019) no. 2, pp. 221-224. doi: 10.21136/MB.2018.0124-17

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