Local Inclusive Distance Vertex Irregular Graphs - Kiki Ariyanti Sugeng; Denny Riama Silaban; Martin Bača; Andrea Semaničová-Feňovčíková (Mathematics, 2021)
Let <semantics>G=(V,E)</semantics> be a simple graph. A vertex labeling <semantics>f:V(G)→{1,2,⋯,k}</semantics> 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 <semantics>x,y∈V(G)</semantics> their weights are distinct, where the weight of a vertex <semantics>x∈V(G)</semantics> 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 <semantics>d=1</semantics> and determine the precise values of the corresponding graph invariant for certain families of graphs.