Unique Isolated Perfect Domination in Graphs

Abstract

A vertex subset S of a graph G is said to be an isolate dominating set(IDS) of G if S is a dominating set and there is at least one isolated vertex in the induced subgraph < S > [8]. An isolated dominating set S of a graph G is called as unique isolate perfect dominating set(UIPDS) of G if there exists exactly one isolated vertex in < S > and the set S is a perfect dominating set. If no proper subset of S is an UIPDS, then S is said to be minima UIPDS. The UIPD number, denoted by  (G) ,is the minimum cardinality of a minimal UIPDS of G. This paper includes some fundamental properties of UIPDS and contains the UIPD number of paths, complete k-partite graphs and disconnected graphs. At the end, the role of UIPDS in the domination chain has been discussed in detail.

 AMS Subject Classification: 05C69