nl-coloring of of vertex-deleted subgraphs

A k-coloring of a simple graph g partitions its vertex set v (g) into k disjoint independent sets. such a coloring is a neighbor-locating coloring if any two vertices within the same color class are distinguished by their external neighborhoods; the minimum such k is the nl-chromatic number, χnl(g). this paper investigates the effect of vertex deletion on this parameter, establishing bounds and exact values for χnl(g − u). to quantify this impact, we introduce the neighbor-locating stability index, ρnl(g), as a measure of a graph’s resilience to structural perturbations. we use this index to classify graph families as absolutely stable, bounded stable, or unstable, analyzing the structural sensitivity of nl-colorings.