Connected certified domination edge critical and stable graphs

Abstract

Abstract In an isolate-free graph 𝒵 = (V 𝒵 , E 𝒵 ), a set C of vertices is termed as a connected certified dominating set of 𝒵 if, |N 𝒵 ( u ) ∩ (V 𝒵 \ C )| = 0 or |N 𝒵 ( u ) ∩ (V 𝒵 \ C )| ≥ 2 ∀u ∈ C , and the subgraph 𝒵[ C ] induced by C is connected. The cardinality of the minimal connected certified dominating set of graph 𝒵 is called the connected certified domination number of 𝒵 denoted by γ cer c ( Z ). In graph 𝒵, if the deletion of any arbitrary edge changes the connected certified domination number, then we call it a connected certified domination edge critical. If the deletion of any random edge does not a ect the connected certified domination number, then we refer to it as a connected certified domination edge stable graph. In this paper, we investigate those graphs which are connected certified domination edge critical and stable upon edge removal. We then study some properties of connected certified domination edge critical and stable graphs.