Split Domination, Independence, and Irredundance in Graphs

In 1978, Kulli and Janakiram \citep{KulliJanakiramSplit} defined the split dominating set: a dominating set $S$ of vertices in a graph $G=(V,E)$ is called {\em split dominating} if the induced subgraph $\langle V \setminus S\rangle$iseitherdisconnectedora$K_1$. In this paper we introduce the properties split independence and split irredundance. A set $S$ of vertices in a graph $G=(V,E)$ is called a {\em split independent set} if $S$ is independent and the induced subgraph $⟨V\setminus S⟩$ is either disconnected or a $K_1$. A set $S$ of vertices in a graph $G=(V,E)$ is called a {\em split irredundant set} if for $u\in S$, $u$ has a private neighbor with respect to $V(S)$ and the induced subgraph $⟨V\setminus S⟩$ is either disconnected or a $K_1$.