STRONG (WEAK) NEIGHBOURHOOD COVERING SETS OF A GRAPH

The ve-degree of a vertex $u \in V(G)$, denoted by $d_{ve}(u)$, is the number of edges in the subgraph $\langle N[u] \rangle$. A vertex $u$ is said to n-cover (neighbourhood-cover) an edge $e$ if $e$ is an edge of the subgraph $\langle N[u] \rangle$. A set $S\subseteq V(G)$ 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 $\alpha_{n}(G)$ 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.