Summary: For a positive integer \(d\), an \(L(d, 1)\)-labeling of a graph \(G\) is an assignment of nonnegative integers to the vertices of \(V(G)\) such that the difference between labels of adjacent vertices is at least \(d\), and the difference between labels of vertices whose distances are two parts is at least 1. The span of an \(L(d, 1)\)-labeling of a graph \(G\) is the difference between the maximum and minimum integers of all labels. The \(L(d, 1)\)-labeling-number of \(G\) is the minimum span over all \(L(d, 1)\)-labelings of \(G\). Based on the work of \(L(d, 1)\)-labels of the edge-path-replacement of a graph \(G\), we study the \(L(2, 1)\)-labeling of the local-edge-path-replacements of the Cartesian products.