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.