Unique isolate domination in graphs

Abstract

A dominating set $S$ of a graph $G$ is said to be an isolate dominating set of $G$ if the induced subgraph $\langle S\rangle$ has at least one isolated vertex [6].
A dominating set $S$ of a graph $G$ is said to be an unique isolate dominating set(UIDS) of $G$ if $\langle S\rangle$ has exactly one isolated vertex. An UIDS $S$ is said to be minimal if no proper subset of $S$ is an UIDS. The minimum cardinality of a minimal UIDS of $G$ is called the UID number, denoted by $\gamma_0^U(G)$. This paper includes some properties of UIDS and gives the UID number of paths, complete $k$-partite graphs and disconnected graphs. Finally, the role played by UIDS in the domination chain has been discussed in detail.