STRONG (WEAK) NEIGHBOURHOOD COVERING SETS OF A GRAPH

The ve-degree of a vertex u ∈ V (G), denoted by d_ve (u), is the number of edges in the subgraph ⟨N[u]⟩. A vertex u is said to n-cover (neighbourhood-cover) an edge e if e is an edge of the subgraph ⟨N[u]⟩. A set S ⊆ V (G) is called a n-covering set of a graph G if every edge in G is n-covered by some vertex in S. The n-covering number α_n (G) is the minimum cardinality of a n-covering set of G. In this paper, we introduce new parameters such as strong (weak) n-covering number and strong (weak) n-independence number using ve-degrees of vertices, and we establish a relationship between them. Further, we define and study n-cover balanced sets.