Local inclusive distance vertex irregular graphs

Let G = (V, E) be a simple graph. A vertex labeling f: V(G) → {1, 2, …, k} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of a graph G if for any two adjacent vertices x, y ∈ V(G) their weights are distinct, where the weight of a vertex x ∈ V(G) is the sum of all labels of vertices whose distance from x is at most d (respectively, at most d but at least 1). The minimum k for which there exists a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of G is called the local inclusive (respectively, non-inclusive) d-distance vertex irregularity strength of G. In this paper, we present several basic results on the local inclusive d-distance vertex irregularity strength for d = 1 and determine the precise values of the corresponding graph invariant for certain families of graphs.