The Split (Nonsplit) Nomatic Number of a Graph

Abstract

For a given connected graph $G = (V, E)$, a set $S \subseteq V(G)$ is a neighborhood set of $G$, if $G = \bigcup\limits_{v\in S}\langle N[v]\rangle$, where $\langle N[v]\rangle$ is the sub graph of $G$ induced by $v$ and all vertices adjacent to $v$. A neighborhood set $S$ is a split (nonsplit) neighborhood set if $\langle V(G)- S \rangle$ is connected (disconnected). The maximum number of a partition of $V(G)$, all of whose are split (nonsplit) neighborhood sets, is the split(nonsplit) nomatic number $N_{s}(G)(N_{ns}(G))$. Our purpose in this paper is to initiate the study of split(nonsplit) nomatic number of a graph. We first study basic properties and bounds for $N_{s}(G)(N_{ns}(G))$. In addition, we determine the $N_{s}(G)(N_{ns}(G))$ of some classes of graphs.