A Comprehensive Analysis of Clique Vertex Neighborhood Numbers

The open neighborhood N(w) of a vertex w ∈ V consists of all vertices adjacent to w in an undirected graph. The closed neighborhood N[w], includes w and all vertices reachable from it. A complete maximal subgraph of G is a clique. A clique k ∈ K(G) cv-covers a vertex v if v ∈ ⟨N[k]⟩, where ⟨N[k]⟩ is the subgraph induced by the closed neighborhood of k. A set S ⊆ K(G) is a cv-neighborhood set if every vertex v is cv-covered by some k ∈ S, that is, G =^⋃ ⟨N[k]⟩. k∈K(G) The minimum cardinality of such a set is the clique vertex neighborhood number n_cv(G). In this paper, we establish bounds for n_cv, characterize graphs attaining these bounds, and compute n_cv for various graph products.