Narrow Results

In a graph $G=(V,E)$, a non-empty set $S\subseteq V(G)$ is said to be an open packing set if no two vertices of $S$ have a common neighbor in $G.$ Let $v\in V$ and let $OS(v)$ denote the maximum cardinality of an open packing set in $G$ which contains $v$. Then $OS(G)=min\{OS(v):v\in V\}$ is called the open packing saturation number of $G$. In this paper, we initiate a study on this parameter.

A set $S\subseteq V(G)$ of a graph $G$ is an $open$$packing$$set$ of $G$ if no two vertices of $S$ have a common neighbor in $G$. An open packing set $S$ is called an outer-connected open packing set(ocop-set) if either $S=V(G)$ or $〈V-S〉$ is connected. The minimum and maximum cardinalities of an ocop-set are called the lower outer-connected open packing number and the outer-connected open packing number, respectively, and are denoted by ${\rho }_{\mathrm{\text{loc}}}^o$ and ${\rho }_{\mathrm{\text{oc}}}^o$, respectively. In this paper, we initiate a study on these parameters.

In a graph $G=(V,E)$, a nonempty set $S\subseteq V$ is said to be an open packing set if no two vertices of $S$ have a common neighbor in $G.$ The maximum cardinality of an open packing set is called the open packing number and is denoted by ${\rho }^o$. In this paper, we examine the effect of ${\rho }^o$ when $G$ is modified by deleting an edge.

In a graph $G=(V,E)$, a set $S\subseteq V(G)$ is said to be an open packing set if no two vertices of $S$ have a common neighbor in $G.$ The maximum cardinality of an open packing set is called the open packing number and is denoted by ${\rho }^o$. The open packing bondage number of a graph $G$, denoted by $ob(G)$, is the cardinality of the smallest set of edges $F\subset E(G)$ such that ${\rho }^o(G-F)>{\rho }^o(G)$. In this paper, we initiate a study on this parameter.

A nonempty set $S\subseteq V(G)$ of a graph $G=(V,E)$ is an open packing set of $G$ if no two vertices of $S$ have a common neighbor in $G$. The maximum cardinality of an open packing set is called the open packing number of $G$ and is denoted by ${\rho }^o(G)$. The open packing subdivision number ${\mathrm{\text{sd}}}_{{\rho }^o}(G)$ is the minimum number of edges in $G$ that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the open packing number. In this paper, we initiate a study on this parameter.