D-irregularity Strength of a Graph

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We initiate to study a \(D\)-irregular labeling, which generalizes both non-inclusive and inclusive \(d\)-distance irregular labeling of graphs. Let \(G=(V(G),E(G))\) be a graph, \(D\) a set of distances, and \(k\) a positive integer. A mapping \(\varphi\) from \(V(G)\) to the set of positive integers \(\{1,2,\dots,k\}\) is called a \(D\)-irregular \(k\)-labeling of \(G\) if every two distinct vertices have distinct weights, where the weight of a vertex \(x\) is defined as the sum of labels of vertices whose distance from \(x\) belongs to \(D\). The least integer \(k\) for which \(G\) admits a \(D\)-irregular labeling is the \(D\)-irregularity strength of \(G\) and denoted by \(\mathrm{s}_D(G)\). In this paper, we establish several fundamental properties on \(D\)-irregularity strength for arbitrary graphs. We also determine this parameter exactly for families of graphs with small diameter or small maximum degree.