Properties of n- Independent Sets and n-Complete Sets of a Graph

The open neighborhood N(w) of a vertex w ∈ V is the set of all vertices that share an edge with w in an undirected graph. In other words, N(w) comprises all vertices that are directly connected to w by an edge, excluding w itself. The enclave of a vertex w ∈ V, denoted as E(w), is the set of all vertices that are reachable from w through a series of adjacent edges, including w itself. In other words, E(w) includes all vertices that can be reached from w by following a path along the edges of the graph. The closed neighborhood of w is known as enclave of w. We note that a vertex w ∈ V, ncovers an edge (Formula presented.), the subgraph induced by the set N[v]. The n-covering number ρn(H) introduced by Sampathkumar and Neeralagi [16] is the minimum number of vertices that n-cover all the edges of H. A set S ⊆ V is said to be n-independent if every edge x ∈ ⟨S⟩ is n-covered by a vertex in V − S. On the other hand, S is n-complete if, for every pair of nonadjacent vertices u, v ∈ S there exists a vertex w ∈ V − S such that {u, v, w} is independent. The n-independence (n-complete) number (Formula presented.) is the maximum order of n-independent (n-complete) set of H. In this paper, a Gallai’s theorem type result (Formula presented.) is proved. In addition to getting several bounds on n-independence number, we show that (Formula presented.) and the chromatic number (Formula presented.).