Concepts Arising from Strong Efficient Domination Number. Part – III

ABSTRACT

Let G = (V, E) be a simple graph. A subset S of V(G) is called a strong (weak) efficient dominating set of G if for every v ∈ V(G), |𝑁𝑠[𝑣] ∩ S|=1. (|𝑁𝑤 [𝑣] ∩ S|=1), where 𝑁𝑠 (𝑣) = {u ∈V(G) : uv ∈ E(G), deg u ≥ deg v}. (𝑁𝑤 (𝑣) = {u ∈V(G) : uv ∈ E(G), deg v ≥ deg u}). The minimum cardinality of a strong (weak) efficient dominating set of G is called the strong (weak) efficient domination number of G and is denoted by 𝛾𝑠𝑒(G) (𝛾𝑤𝑒(G)). A graph G is strong efficient if there exists a strong efficient dominating set of G. The strong efficient co-bondage number 𝑏𝑐𝑠𝑒(G) is the maximum cardinality of all sets of edges X ⊆ E such that 𝛾𝑠𝑒(𝐺 + 𝑋) ≤ 𝛾𝑠𝑒(G). In this paper, further results on strong efficient co-bondage number of some special graphs are determined.